In the figure, if \(\triangle A B E \cong \triangle \text { ACD, show that }\triangle A D E \sim \triangle A B C\)

Prove that the area of an equilateral triangle described on one side of a square in equal to half  the area of the equilateral triangle described on one of its diagonals.

Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are \(1 \frac{1}{2}\) times the corresponding sides of the isosceles triangle.

O is any point inside a triangle \(\Delta\)ABC The bisector of \(\angle A O B, \angle B O C, \angle C O A\) meet the sides AB, BC and CA in point D, E and F respectively. Show that AD x BE x CF = DB x EC x FA.

In AD is the median of  \(\Delta\)ABC The bisector of \(\angle\)ADB and \(\angle\)ADC meet AB and AC in E and F respectively. Prove that EF || BC

The bisectors of the angle B and C of a triangle ABC, meet the opposite sides in D and E respectively. If DE DE || BC BC, prove that the triangle is isosceles

P and Q are points on the sides CA and CB respectively of \(\Delta\)ABC right angled at C. Prove that AC² + BP² = AB² + PQ².

ABC is a right triangle right angled at C and \(\mathrm{AC}=\sqrt{3} \mathrm{BC} \text {. Prove that } \angle A B C=60^{\circ} \text {. }\)

A point O in the interior of a rectangle ABCD is joined with each of the vertices A, B, C and D. Prove that OB² + OD² = OC² + OA²

Show that the alternates of a triangle are concurrent.

Draw a circle of radius 5 cm. Consider a point P at a distance of 8 cm from the circle's centre and draw two tangents from P.

Construct a ​\(\triangle\)​ PQR in which PQ = 8 cm∠" id="MathJax-Element-90-Frame" role="presentation" style="position: relative;" tab index="0"> <∠R = 60^o and the median RG from R to PQ is 5.8 cm. Find the length of the altitude from R to PQ.