Exam Practice

In a $△ABC$, if $\angle A=60°$ and the altitudes from $B$ and $C$ meet $AC$ and $AB$ at $P$ and $Q_1$. fespectively and intersect each other at $I$.

Find the measure of $\angle BIC$.

Two equal chords $AB$ and $CD$ of a circle with centre $O$, when produced, meet at a point $E$ as shown in figure. Prove that $BE=DE$ and $AE=CE$.

In the following figure, $O$ is the centre of the circle and $\angle BCO=30°$. Find $x$ and $y$,

If bisectors of opposite angles of a cyclic quadrilateral $ABCD$ intersect the circle circumscribing it at the points $P$ and $Q$, then prove that $PQ$ is a diameter of the circle.

$A, B$ and $C$ are three points on a circle. Prove that the perpendicular bisector of $AB, BC$ and $CA$ are concurrent.

$AB$ and $AC$ are two chords of a circle of radius $r$ such that $AB=2AC$. If $p$ and $q$ are the distances of $AB$ and $AC$ from the centre, then prove that $4q^2=4p^2+3r^2$,

Prove that the mid-point of the hypotenuse of a right angled triangle is equidistant from its vertices.

Prove that any angle subtended by a minor arc in the alternate segment is acute and any angle subtended by a major arc in the alternate segment is obtuse.