Thread: geometry - pentagons

In a pentagon ABCDE,

AB = AE

BC = ED

BD = EC

Prove that AC = AD

Hint:
Block your pentagon with a circle.

Originally Posted by Also sprach Zarathustra

Hint:
Block your pentagon with a circle.

My mistake! Sorry!

Originally Posted by the undertaker

In a pentagon ABCDE,

AB = AE

BC = ED

BD = EC

Prove that AC = AD

(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.