Properties of Triangle

ABC is an isosceles triangle with an inscribed circle with centre O. Let P be the midpoint of BC. If AB = AC = 15 and BC = 10, then OP equals

The angles of a triangle are in the ratio 2 : 3 : 7 and the radius of the circumscribed circle is 10 cm. The length of the smallest side is

The three sides of a right angled triangle are in GP (geometric progression). If the two acute angles be \(\alpha\) and \(\beta\), then tan\(\alpha\) and tan\(\beta\) are

If in a \(\Delta\)ABC, AD, BE and CF are the altitudes and R is the circumradius, then the radius of the circumcircle of \(\Delta\)DEF is

Let p, q and r be the altitudes of a triangle with area S and perimeter 2t. Then, the value of \({1 \over p} + {1 \over q} + {1 \over r}\) is

If cos A + cos B + cos C = 0, prove that cos 3A + cos 3B + cos 3C = 12 cos A cos B cos C.

If in a triangle ABC, sin A, sin B, sin C are in A.P., then

In a \(\Delta\)ABC, \(2ac\sin \left( {{{A - B + C} \over 2}} \right)\) is equal to

If \({{\cos A} \over 3} = {{\cos B} \over 4} = {1 \over 5}, - {\pi \over 2} < A < 0, - {\pi \over 2} < B < 0\), then value of \(2\sin A + 4\sin B\) is

In a right-angled triangle, the sides are a, b and c, with c as hypotenuse, and c \(-\) b \(\ne\) 1, c + b \(\ne\) 1. Then the value of \(({\log _{c + b}}a + {\log _{c - b}}a)/(2{\log _{c + b}}a \times {\log _{c - b}}a)\) will be

In a triangle PQR, \(\angle\)R = \(\pi\)/2. If \(\tan \left( {{P \over 2}} \right)\) and \(\tan \left( {{Q \over 2}} \right)\) are roots of ax2 + bx + c = 0, where a \(\ne\) 0, then which one is true?

In triangle ABC, if \(\sin A\sin B = {{ab} \over {{c^2}}}\), then the triangle is

If a = 2\(\sqrt2\), b = 6, A = 45\(^\circ\), then