Some Properties of a Triangle

If one of the three angles of the equilateral triangle measures $60°$, then find the other two angles.

All the sides of an equilateral triangle are

All the sides of an equilateral triangle are equal.

All the three angles of the equilateral triangle have the same measure.

Prove that equilateral triangle is equiangular.

In an isosceles triangle $\mathrm{A}\mathrm{B}\mathrm{C},$ with $\mathrm{A}\mathrm{B}=\mathrm{A}\mathrm{C},$ the bisectors of $\mathrm{\angle }\mathrm{B}$ and $\mathrm{\angle }\mathrm{C}$ intersect each other at $\mathrm{O}.$ Join $\mathrm{A}$ to $\mathrm{O}.$ Is $\mathrm{O}\mathrm{B}=\mathrm{O}\mathrm{C}$ (Yes/No).

In the figure, $\mathrm{PA}$ is a tangent to the circle, $\mathrm{PBC}$ is secant and $\mathrm{AD}$ bisects angle $\mathrm{BAC}.$ Show that triangle $\mathrm{PAD}$ is an isosceles triangle. Also, show that $\angle \mathrm{CAD}=\frac{1}{2} [\angle \mathrm{PBA}-\angle \mathrm{PAB}].$

Tangent at $\mathrm{P}$ to the circumcircle of triangle $\mathrm{PQR}$ is drawn. If this tangent is parallel to side $\mathrm{QR}$, show that $△\mathrm{PQR}$ is isosceles.
 

$\mathrm{AB}$ is the diameter and $\mathrm{AC}$ is a chord of a circle with centre $\mathrm{O}$ such that angle $\mathrm{BAC}=30°.$ The tangent to the circle at $\mathrm{C}$ intersects $\mathrm{AB}$ produced in $\mathrm{D}.$ Show that $\mathrm{BC}=\mathrm{BD}.$
 

In $\Delta ABC$ if $AB=BC$, then

In $△\mathrm{P}\mathrm{Q}\mathrm{R}, \mathrm{\angle }\mathrm{R}=\mathrm{\angle }\mathrm{P}$ and $\mathrm{Q}\mathrm{R}=4 \mathrm{c}\mathrm{m}$ and $\mathrm{P}\mathrm{R}=5 \mathrm{c}\mathrm{m}$. Then the length of $PQ$ is

The circumcentre of the triangle, whose vertices are $\left(0,0\right), \left(3,\sqrt{3}\right)$ and $\left(0,2\sqrt{3}\right)$, is

Let $(-4,3)$ and $(4,3)$ be two vertices of an equilateral triangle and the origin lies in the interior of the triangle. Then, the coordinates of the third vertex are