Given triangle ABC, pick any point P that lies in its interior. Th...

Question
Given triangle ABC, pick any point P that lies in its interior. Through P, construct the three lines parallel to the sides of the triangle, as shown.
The lengths of the segments in bold are labeled.
Prove: (x/BC)+(y/AC)+(z/AB)=1

[Hint: The problem with these proofs is that its not exactly clear where to start. There are lots of similar triangles in the diagram, but which ones do we need? Well, look at the equation you need to prove; the segment lengths involved here are from the three middle triangles and the big triangle. Are any of these triangles similar? Why? Also, in whatever problem you happen to be working on from now on, you should be looking for our old friends lurking about; for example, see the parallelograms? and as usual, write equations, use algebra]
No angle chasing is necessary. Fiddle with angles to get similar triangles.
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There are many parallel lines in this triangle that should allow us to use properties of similar triangles. This is a tricky problem and it is easy to get confused.

We need to show that: x/BC + y/AC+z/AB=1.

Let us take a look at the quantity 1- z/AB. We need to show that it is equal to x/BC + y/AC.
Now 1-z/AB=(AB-z)/AB.
And AB-z is the length of two line segment, one from A to the line segment z, that I will call AZ
and the other ZB.
So (AB-z)/AB = AZ/AB + ZB/AB.
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      • Illuminate\Cache\Events\CacheMissed (352μs)
      • Illuminate\Database\Events\StatementPrepared (631μs)
      • Illuminate\Database\Events\QueryExecuted (935μs)
      • eloquent.retrieved: App\Models\SubjectCat (91μs)
      • Illuminate\Cache\Events\KeyWritten (284μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (266μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • Illuminate\Cache\Events\CacheHit (410μs)
      • Illuminate\Cache\Events\CacheHit (273μs)
      • Illuminate\Cache\Events\CacheHit (431μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (166μs)
      • Illuminate\Cache\Events\CacheHit (328μs)
      • Illuminate\Cache\Events\CacheHit (274μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (423μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (170μs)
      • creating: site.layouts.app (459μs)
      • composing: site.layouts.app (24μs)
      • creating: components.canonical (370μs)
      • composing: components.canonical (88μs)
      • creating: components.open-graph (166μs)
      • composing: components.open-graph (80μs)
      • creating: site.headers.header (337μs)
      • composing: site.headers.header (82μs)
      • Illuminate\Cache\Events\CacheHit (2.87ms)
      • creating: components.footer (84μs)
      • composing: components.footer (94μs)
      • Illuminate\Cache\Events\CacheHit (767μs)
      • Illuminate\Cache\Events\CacheMissed (268μs)
      • Illuminate\Database\Events\StatementPrepared (535μs)
      • Illuminate\Database\Events\QueryExecuted (1.03ms)
      • eloquent.retrieved: App\Models\SubjectCat (210μs)
      • Illuminate\Cache\Events\KeyWritten (862μs)
      • Illuminate\Cache\Events\CacheHit (285μs)
      • Illuminate\Cache\Events\CacheHit (269μs)
      • Illuminate\Cache\Events\CacheHit (186μs)
      • Illuminate\Cache\Events\CacheHit (175μs)
      • Illuminate\Cache\Events\CacheHit (249μs)
      • Illuminate\Cache\Events\CacheHit (307μs)
      • Illuminate\Cache\Events\CacheHit (409μs)
      • Illuminate\Cache\Events\CacheHit (189μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (306μs)
      • Illuminate\Cache\Events\CacheHit (374μs)
      • creating: components.forms.contact-us (225μs)
      • composing: components.forms.contact-us (156μs)
      • creating: components.forms.get-started (111μs)
      • composing: components.forms.get-started (65μs)
      • creating: components.forms.free-tool-download (101μs)
      • composing: components.forms.free-tool-download (63μs)
      • creating: components.forms.claim-free-worksheet (98μs)
      • composing: components.forms.claim-free-worksheet (63μs)
      • creating: components.forms.tutor-subscription-waitlist (96μs)
      • composing: components.forms.tutor-subscription-waitlist (63μs)
      • creating: components.forms.tutor-subscription-join (91μs)
      • composing: components.forms.tutor-subscription-join (77μs)
      • creating: components.forms.tutor-support (103μs)
      • composing: components.forms.tutor-support (127μs)
      • 311 x Illuminate\Cache\Events\CacheHit (14.68%)
        99.85ms
        20 x Illuminate\Database\Events\StatementPrepared (6.06%)
        41.22ms
        20 x Illuminate\Database\Events\QueryExecuted (4.52%)
        30.77ms
        11 x Illuminate\Cache\Events\CacheMissed (1.61%)
        10.95ms
        11 x Illuminate\Cache\Events\KeyWritten (1.21%)
        8.27ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.76%)
        5.14ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.39%)
        2.68ms
        1 x Illuminate\Routing\Events\Routing (0.27%)
        1.83ms
        12 x eloquent.retrieved: App\Models\SubjectCat (0.22%)
        1.51ms
        1 x Illuminate\Routing\Events\RouteMatched (0.08%)
        543μs
        1 x eloquent.booting: App\Models\SubjectCat (0.08%)
        538μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.08%)
        516μs
        1 x creating: site.layouts.app (0.07%)
        459μs
        6 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.07%)
        456μs
        1 x creating: components.canonical (0.05%)
        370μs
        1 x creating: homework.show (0.05%)
        369μs
        1 x creating: components.breadcrumbs (0.05%)
        353μs
        1 x creating: site.headers.header (0.05%)
        337μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        300μs
        1 x eloquent.booting: App\Models\Subject (0.04%)
        291μs
        2 x eloquent.retrieved: App\Models\Subject (0.04%)
        269μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        265μs
        1 x creating: components.forms.contact-us (0.03%)
        225μs
        1 x composing: components.breadcrumbs (0.03%)
        172μs
        1 x creating: components.open-graph (0.02%)
        166μs
        1 x composing: homework.show (0.02%)
        165μs
        1 x composing: components.forms.contact-us (0.02%)
        156μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        153μs
        1 x composing: components.forms.tutor-support (0.02%)
        127μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        123μs
        1 x creating: components.forms.get-started (0.02%)
        111μs
        1 x creating: components.forms.tutor-support (0.02%)
        103μs
        1 x creating: components.forms.free-tool-download (0.01%)
        101μs
        1 x creating: components.forms.claim-free-worksheet (0.01%)
        98μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        97μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.01%)
        96μs
        1 x composing: components.footer (0.01%)
        94μs
        1 x creating: components.forms.tutor-subscription-join (0.01%)
        91μs
        1 x composing: components.canonical (0.01%)
        88μs
        1 x creating: components.footer (0.01%)
        84μs
        1 x composing: site.headers.header (0.01%)
        82μs
        1 x composing: components.open-graph (0.01%)
        80μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        77μs
        1 x composing: components.forms.get-started (0.01%)
        65μs
        1 x composing: components.forms.free-tool-download (0.01%)
        63μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        63μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        63μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        63μs
        1 x composing: site.layouts.app (0%)
        24μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      20 statements were executed, 5 of which were duplicates, 15 unique. Show only duplicated50.84ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '11249' limit 1
        20.33mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 11249
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (3)
        1.05mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (11249)
        3.37mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.27mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 11249 and `subject` = 3 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        3.33mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 11249
        • 1: 3
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (3)
        940μstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 11249 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.65mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 11249
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 11249 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.15mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 11249
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 11249 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        1.04mstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.12mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.17mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1.07mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        3.29mstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        990μstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        930μstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        4.66mstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        910μstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        860μstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        860μstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        850μstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
      12SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      6HomeworkLibraryFile.php
      App\Models\Subject
      2Subject.php
          _token
          VL3itrucY591HGLtqZQGeBVQvVYHyAPWYt4gEbss
          utm_source
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          redirectUrl
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          PHPDEBUGBAR_STACK_DATA
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          status_code
          200
          
          status_text
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          format
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          content_type
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          request_query
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          request_request
          []
          
          request_headers
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          request_cookies
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