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

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

Details: Click on the figure below.

## Saturday, February 10, 2018

### Geometry Problem 1355: Triangle, Midpoint, Median, Circle, Chord, Equal Product of measure of segments

Labels:
chord,
circle,
geometry problem,
measurement,
median,
midpoint,
product,
triangle

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https://photos.app.goo.gl/h4PcQ74of1KJy3eh2

ReplyDeleteDraw AP//DE and CN//DE

See sketch for positions of points P,Q, N

Triangle BAP is isosceles => BP=BA

Triangles AQM and CNM are congruent ( case ASA) => AQ=NC

Since DE//AP => GE/DG=PQ/AQ=PQ/NC

Triangle BPQ similar to BCN => PQ/NC= BP/BC=AB/BC => GE/DG=AB/BC

Or GE . BC= DG . AB

Draw perpendiculars h1 and h2 from G to AB and BC

ReplyDeleteSince S (ABG) = S (CBG),

AB.h1 = BC.h2..... (1)

Since BD = BE,

GE.h1 = DG.h2..... (2)

Divide (1) by (2) and the result follows

Sumith Peiris

Moratuwa

Sri Lanka

Using areas it is straight forward.

ReplyDeleteBest regards

Let BM split angle ABC into angles x and y.

ReplyDeleteArea of triangle ABM = Area of triangleBCM.

So ½ AB.MB.sin x= ½ BC.MB.sin y,

Implies AB/BC = sin y/sin x .

Next EG/DG = (BGE)/(BGD)

= ½ BE.BG sin y / ½ BD.BG sin x

= sin y/sin x

Hence AB/BC = EG/DG,

AB.DG = BC.EG

Vijaya Prasad Nalluri (Pravin)

Rajahmundry - INDIA