### Video Transcript

The points π΄, π΅, and πΆ all lie
on the circumference of a circle with centre π. The line π΄πΆ passes through the
point π. Prove that angle π΄π΅πΆ is right
angled.

In order to solve this problem, we
need to use our angle properties and our properties of circles. Firstly, we know that the length
ππ΄ is equal to the length ππ΅ and also the length ππΆ as they all radii of the
circle. As ππ΄ is equal to ππ΅, triangle
ππ΄π΅ is isosceles. In the same way because ππ΅ is
equal to ππΆ, ππ΅πΆ is isosceles.

As these triangles are isosceles,
we know that angle ππ΅πΆ is equal to angle ππΆπ΅. We have defined this as π on our
diagram. Likewise, angle ππ΄π΅ is equal to
angle ππ΅π΄. We have defined this as π on the
diagram. If we define angle π΄ππ΅ as the
Greek letter π and angle π΅ππΆ as the Greek letter π, we know that π plus π
equals 180 degrees as the sum of two angles on a straight line equals 180
degrees.

We also know that angles in a
triangle add up to 180 degrees. If we consider the orange triangle
ππ΄π΅, we can see that two π plus π equals 180. In the same way, in the pink
triangle, two π plus π equals 180. Subtracting π from both sides of
equation one gives us two π equals 180 minus π and dividing both sides of this
equation by two gives us π is equal to 180 minus π divided by two. We can rearrange equation two in
the same way. Two π is equal to 180 minus π and
π is equal to 180 minus π divided by two.

The aim of our question is to prove
that angle π΄π΅πΆ is right angled. Angle π΄π΅πΆ is equal to π plus
π. Substituting in our expressions for
π and π gives us 180 minus π divided by two plus 180 minus π divided by two. As we have a common denominator of
two, we can rewrite this as 360 minus π minus π divided by two. Factorizing out a negative one from
the last two terms on the top gives us 360 minus π plus π all divided by two.

We know that π plus π is equal to
180 degrees as they are two angles on a straight line. We, therefore, have 360 minus 180
divided by two. 360 minus 180 is 180. Dividing this by two gives us an
answer of 90 degrees. We have, therefore, proved that
angle π΄π΅πΆ is a right angle as it is equal to 90 degrees.

This is actually the proof of one
of our circle theorems. Any angle in a semicircle or from a
diameter is equal to 90 degrees. As π΄πΆ is a diameter of a circle,
angle π΄π·πΆ would be 90 degrees and angle π΄πΈπΆ would also be 90 degrees.