Question

ABCD is a cyclic quardrilateral in which arc AD = arc DC. BC is produced to E such way that AB = CE than prove that triangle DBE is an isosceles triangle.

Answer 1

In a cyclic quadrilateral, the sum of the opposite angles is 180 degrees.

Inscribed angles that intercept the same arc are equal.

ABCD is a cyclic quadrilateral in which arc AD = arc DC. BC is produced to E in such a way that AB = CE.

Since ABCD is a cyclic quadrilateral, the sum of the opposite angles is 180 degrees. So angle ADC is equal to angle ABC.

the arc AD is equal to the arc DC, the angle ADC and the angle ABC intercept the same arc. So angle ADC is equal to angle ABC.

angle ADC is equal to angle ABC, and also that angle ADC is equal to angle ABC.

conclusion: the angle ABC is equal to the angle ADC, and therefore the triangle DBE is an isosceles triangle. Sides DB and BE have the same length, making triangle DBE an isosceles triangle.

Explanation: