Q24 of 141 Page 438

In the given figure, is a quadrilateral in which and Show that the points lie on a circle.

In ΔADE and ΔBCF,


AD = BC


∠AED = ∠BFC


∠ADE = ∠BCF [∠ADC - 90° = ∠BCD - 90°]


∴ ΔADE ≅ ΔBCF


The Cross ponding parts of the congruent triangles are equal.


∴ ∠A = ∠B


Now,


∠A + ∠B + ∠C + ∠D = 360°


⇒ 2 ∠B + 2 ∠D = 360°


⇒ ∠B + ∠D = 180°


∴ ABCD is a cyclic quadrilateral.


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Give a geometrical construction for finding the fourth point lying on a circle passing through three given points, without finding the center of the circle. Justify the construction.

 23

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26

The diagonals of a cyclic quadrilateral are at right angles. Prove that the perpendicular from the point of their intersection on any side when produced backwards, bisects the opposite side.