Two triangles ABC and PQR are such that the perpendiculars from A to QR ,B to RP and C to PQ are concurrent .Show that the perpendicular from P to BC ,Q to CA and R to AB are also concurrent .

Show that the perpendicular bisectors of the sides of a triangle are concurrent.

A triangle ABC has angle B = angle C . Prove that : (i) the perpendiculars from the mid-point of BC to AB and AC are equal. (ii) the perpendicular from B and C to the opposite sides are equal.

Prove that the perpendicular let fall from the vertices of a triangle to the opposite sides are concurrent.

P is any point in the angle ABC such that the perpendicular drawn from P on AB and BC are equal. Prove that BP bisects angle ABC.

Two coplanar triangles ABC and PQR re such that the lines drawn through the vertices of /_\ABC parallel to the sides of /_\PQR are concurrent. Prove that the lines drawn through the vertices of /_\PQR parallel to sides of /_\ABC are concurrent.

From a point O inside a triangle A B C , perpendiculars O D ,O Ea n dOf are drawn to rthe sides B C ,C Aa n dA B , respecrtively. Prove that the perpendiculars from A ,B ,a n dC to the sides E F ,F Da n dD E are concurrent.

ABC is an isosceles triangle with AB = AC =13 cm and BC = 10 cm. Calculate the length of the perpendicular from A to BC.

In a triangle ABC, AB = BC, AD is perpendicular to side BC and CE is perpendicular to side AB. Prove that : AD = CE.

Perpendicular bisectors of the sides AB and AC of a triangle ABC meet at O. Does the perpendicular bisector of BC pass through O?

The vertices of a triangle ABC are the points (0, b), (-a, 0), (a, 0) . Find the locus of a point P which moves inside the triangle such that the product of perpendiculars from P to AB and AC is equal to the square of the perpendicular to BC .