In a pentagon ABCDE,

AB = AE

BC = ED

BD = EC

Prove that AC = AD

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- June 24th 2010, 02:21 AMthe undertakergeometry - pentagons
In a pentagon ABCDE,

AB = AE

BC = ED

BD = EC

Prove that AC = AD - June 24th 2010, 03:16 AMAlso sprach Zarathustra
Hint:

Block your pentagon with a circle. - June 24th 2010, 03:29 AMAlso sprach Zarathustra
- June 24th 2010, 05:39 AMmathaddict
(1) Triangle EDC is congruent to triangle BCD (SSS).

(2) Since angles in the same segment are equal(angle DEC = angle DBC), we can deduce that this pentagon is inscribed in a circle.

(3) Chords AB=AE implies arc AB=AE , angle,x subtended by these arcs are the same. Similarly, BC=ED, angle,y subtended will be the same too.

angle ADC=angle ACD= x+y and since base angles are equal, triangle ADC is an isosceles triangle.