### Video Transcript

π΄, π΅, and πΆ are points on a
circle and π is the center of the circle. π·πΆπΈ is a tangent to the
circle. πΆπ΅ bisects the angle π΄πΆπΈ. Angle π΅πΆπΈ is equal to π₯ degrees
and angle ππ΅πΆ is equal to 90 minus π₯ degrees. Prove that π΄π΅ is equal to
π΅πΆ.

In order to prove that π΄π΅ is
equal to π΅πΆ, we need to show that triangle π΄π΅πΆ is isosceles. If this is the case, it will have
two equal angles at π΅π΄πΆ and π΅πΆπ΄. And this will mean the sides π΄π΅
and π΅πΆ have to be equal in length. Letβs first see if thereβs any
useful information that we can use in the question itself.

Weβre told that πΆπ΅ bisects the
angle π΄πΆπΈ; that is to say, the line πΆπ΅ cuts the angle π΄πΆπΈ in half. For this to be the case, that must
mean that angle π΅πΆπΈ is equal to angle π΅πΆπ΄. Theyβre both π₯ degrees. Next, we need to recall any circle
theorems that might help us here. Straightaway, I can see one that
might be useful. We know that two radii or radiuses
form an isosceles triangle. ππ΅ and ππΆ are lines which join
the center of the circle to a point on its circumference. That means theyβre the radii of the
circle and triangle ππ΅πΆ is isosceles.

This means we know the ππΆπ΅ and
ππ΅πΆ are equal. Theyβre both 90 minus π₯
degrees. We also know that angles in a
triangle sum to 180 degrees. So we can use this information to
calculate the size of angle π΅ππΆ marked. Angle π΅ππΆ can be found by
subtracting angle ππ΅πΆ and angle ππΆπ΅ from 180 degrees. 180 minus 90 minus 90 is zero. Negative negative π₯ is simply π₯,
minus negative π₯ is plus π₯. And we can see that angle π΅ππΆ is
equal to two π₯ degrees.

We can see now that angle π΅ππΆ
and π΅π΄πΆ are subtended from points on the circumference of the circle. We can, therefore, use the circle
theorem that says the angle at the center is twice the angle at the
circumference. Another way to think about this is
that the angle at the circumference is half the angle at the center. Angle π΅π΄πΆ is half angle
π΅ππΆ. Itβs a half of two π₯ which is
simply π₯ degrees.

We have now shown that π΄π΅πΆ is an
isosceles triangle with equal angles at π΅π΄πΆ and π΅πΆπ΄. This means in turn that the sides
π΄π΅ and π΅πΆ must also be of equal length. We have proven that π΄π΅ is equal
to π΅πΆ.