perplexus.info :: Geometry : Circumcenters in trapezoid

Circumcenters in trapezoid (Posted on 2025-03-11)

The trapezoid ABCD of bases AB and CD, has angle A = 90 degrees, AB = 6, CD = 3 and AD = 4. Let E, G, H be the circumcenters of triangles ABC, ACD, ABD, respectively. Find the area of the triangle EGH.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

| Comment 1 of 2

Let F be the midpoint of AB.

The trapezoid is then composed of three 3-4-5 right triangles: CDA, AFC, BFC.

G is the midpoint of AC, H is the midpoint of BD. These points are midway between the trapezoid bases and 3/2 units apart.

E is on line FC (as is H) a distance 7/8 from F. EH=9/8

Thus triangle EGH is a right triangle with area (1/2)(3/2)(9/8)=27/32

Posted by Jer on 2025-03-11 12:11:38

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