Question

The diagonals of a parallelogram ABCD intersect in appoint E. Show that the circumcircles ∆ADE and ∆BCE touch each other at E.

Answer 1

The diagonals of a parallelogram ABCD intersect in appoint E. Show that the circumcircles ∆ADE and ∆BCE touch each other at E.

Here is a solution

Answer 2

Parallelogram = ABCD (Given)

Intersection point = E  (Given)

∠ AEX = ∠ ADB ( Alternate segment theorum)

∠ ADB = ∠ CBD ( As alternate interior angles are equal)

∠ AEX = ∠ CEY ( Vertically opposite angles)

Therefore,

∠ CBD = ∠ CEY

Thus, as per the converse of alternate segment theorum, l will be tangent to the second circle with point of contact as E.

Therefore, the circumcircles ∆ADE and ∆BCE touch each other at point E.