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Inscribe Right Triangle (Posted on 2010-07-18) Difficulty: 3 of 5

Let Q be a point on side BC of triangle ABC.

Construct points P and R on sides AB and AC
respectively such that PQR is a right angle
and PR is parallel to BC.

  Submitted by Bractals    
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Solution: (Hide)

Construct point O as the midpoint of side BC.
Construct circle K having BC as a diameter.
Ray AQ intersects circle K at point D with Q
between A and D. Join O wth A. Construct a
line through Q parallel to OD intersecting OA
at point E. Construct a line through E parallel
to BC intersecting sides AB and AC in points
P and R respectively. Join P with Q and Q
with R.

Proof: 

Similar triangles AER ~ AOC and AOD ~ AEQ:

    |AE|     |AO|     |AO|     |AE| 
   ------ = ------ = ------ = ------.
    |ER|     |OC|     |OD|     |EQ|

Therefore,

   |ER| = |EQ|.

Similar triangles AEP ~ AOB and AOD ~ AEQ:

    |AE|     |AO|     |AO|     |AE| 
   ------ = ------ = ------ = ------.
    |EP|     |OB|     |OD|     |EQ|

Therefore,

   |EP| = |EQ| = |ER|.

Therefore, Q lies on a circle with PR
as a diameter. Thus, PQR is a right angle.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some ThoughtsThoughtsBrian Smith2010-07-20 15:39:58
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