Question

In the given figure, altitudes YZ and XT of ∆WXY intersect at P. Prove that,

1. `square`WZPT is cyclic.
2. Points X, Z, T, Y are concyclic.

Answer 1

(1) ∠WZP = ∠WTP = 90°

∴ ∠WZP + ∠WTP = 90° + 90° = 180°

∴ by converse of cyclic quadrilateral theorem,

`square`WZPT is cyclic.

(2) ∠XZY = ∠XTY = 90°   ...(Given)

∴ points X and Y on line XY subtend equal angles at two distinct points Z and T on the same side of line XY.

If two points on a given line subtend equal angles at two distinct points that lie on the same side of the line, then the four points are concyclic.

∴ Points X, Z, T, Y are concyclic.