Properties of Triangles

$A$ and $B$ are two points on one bank of a straight river and $C, D$ are two other points on the other bank of river, if direction from $A$ to $B$ is same as that from $C$ to $D$ and $A B=a, \angle C A D=\alpha, \angle…

If an isosceles triangle of vertical angle $2 \theta$ is inscribed in a circle of radius $a$. Then, area of the triangle is maximum, when $\theta$ is equal to

In a triangle, the lengths of two larger sides are 10 cm and 9 cr. . If the angles of the triangle are in AP , then the length of the third side is

The $\triangle P Q R$ is inscribed in the circle $x^2+y^2=25$. If $Q$ and $R$ have coordinates $(3,4)$ and $(-4,3)$, respectively. Then, $\angle Q P R$ is equal to

If in a $\triangle P Q R, \sin P ; \sin Q, \sin R$ are in $A P$, then

In $\triangle A B C$, if $a=30, b=24,c=18$, then $r_3$ is equal to

In $\triangle A B C, \frac{b-c}{r_1}+\frac{c-a}{r_2}+\frac{a-b}{r_3}$ is equal to

In a $\triangle A B C$, if $\sin A \sin B=\frac{a b}{c^2}$, then the triangle is

The equation of the circle circumscribing the triangle formed by the lines $x+y=6$, $2 x+y=4$ and $x+2 y=5$ is

In an equilateral triangle, the in-radius, circum-radius and one of the ex-radii are in the ratio

In $\triangle A B C, \frac{b-c}{r_1}+\frac{c-a}{r_2}+\frac{a-b}{r_3}$ is equal to

In a $\triangle A B C$, if $b=2, \angle B=30^{\circ}$, then the area of the circumcircle of $\triangle A B C$ (in sq units) is