Geometry Problem. Post your solution in the comment box below.

Level: Mathematics Education, High School, Honors Geometry, College.

Details: Click on the figure below.

## Wednesday, October 24, 2018

### Geometry Problem 1395: Triangle with three rectangles on the sides, Midpoints, Perpendicular lines, Concurrency

Labels:
concurrency,
concurrent,
geometry problem,
midpoint,
perpendicular,
rectangle,
side,
triangle

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https://photos.app.goo.gl/3a76E7ztEMNkD1fw6

ReplyDeleteLet u, v, x, y , w, z are angles of triangles HCF, BDG and AEJ as shown in the sketch

In triangle HCF draw altitude Q’Ct

Note that ∠ (BCt)= u and ∠ (ACt)= v ( angles formed by perpendicular lines)

We also have ∠ (MKQ)= ∠ (BCt)= u ( angles formed by parallel lines)

And ∠ (LKQ)= ∠ (ACt)= v

Similarly we also have x, y,w, z inside triangle KLM as shown

We have sin(u)/sin(v)= CH/CF , sin(y)/sin(x)= BD/BG and sin(w)/sin(z)= AJ/AE

Multiply 3 expressions side by side and note that ABDE, BCFG and ACHJ are the rectangles

We have sin(u)/sin(v) * sin ( y)/sin(x) * sin(w)/sin(z)= 1

So KQ , NL and MP are concurrent per Ceva’s theorem