Chapter: Circle· Mathematics Part 2 (Geometry)

In figure 3.102, two circles intersect each other at points A and E. Their common secant through E intersects the circles at points B and D. The tangents at B and D intersect at C. Prove that ABCD is cyclic.

In figure 3.102, two circles intersect each other at points A and E. Their common secant through E intersects the circles at points B and D. The tangents of the circles at points B and D intersect each other at point C. Prove that \(\square ABCD\) is cyclic.

[Figure: Two circles intersect at A (upper) and E (lower). Line B–E–D: B is on the left (smaller) circle and D on the right (bigger) circle. The tangent to the left circle at B and the tangent to the right circle at D meet at C below the line BD.]

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Construction: Draw seg AB, seg AE and seg AD.

Proof:

Line CB is a tangent to the left circle at B and seg BE is a chord.

\(\therefore \angle CBE = \angle BAE\) ...(1) (tangent secant angle theorem)

Similarly, line CD is a tangent to the right circle at D and seg DE is a chord.

\(\therefore \angle CDE = \angle DAE\) ...(2) (tangent secant angle theorem)

In \(\Delta BCD\), \(\angle CBD + \angle CDB + \angle BCD = 180^\circ\) (sum of angles of a triangle)

\(\therefore \angle CBE + \angle CDE + \angle BCD = 180^\circ\) (B – E – D)

\(\therefore \angle BAE + \angle DAE + \angle BCD = 180^\circ\) ...[from (1) and (2)]

\(\therefore \angle BAD + \angle BCD = 180^\circ\) (angle addition property)

Thus a pair of opposite angles of \(\square ABCD\) is supplementary.

\(\therefore \square ABCD\) is cyclic.

Doubtly AI · 4 Nov 2026 AI-verified · Nov 2026 Report

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In figure 3.102, two circles intersect each other at points A and E. Their common secant through E intersects the circles at points B and D. The tangents at B and D intersect at C. Prove that ABCD is cyclic.

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