To prove the theorem, we need to show that 2 (a + b) = c. We will label each angle, as is shown below. We will also have two angles in each isosceles that are the same. This gives us two isosceles triangles with two sides of length r, and two sides of the same length. Let us construct the same shape, but now also construct a line from the 'x' point to the centre. The angle at the centre is double the angle at the circumference, Ben Cairns-Vaia Proof of theorem 2: The angle at the centre is double the angle at the circumference If this happens then the theorem will still hold. What is important to note is that it doesn't matter where the point is on the arc, as long as it is between the two unmarked angles. Here, the angle subtended by an arc at the centre is twice the angle subtended at the circumference. QED Theorem 2: The angle at the centre is double the angle at the circumference The angle at the circumference is given by x + y, and thus, the angle is right-angled. As 2x + 2y = 180 °, it follows – by dividing by two – that x + y = 90 °. Looking at the largest triangle, we know that 2x + 2y = 180 ° as the angles must sum to 180 °. This means each triangle has two angles which are the same. This will mean we have divided the triangle into two further triangles, each isosceles, with two sides of length r (we use r to denote radius length). Let us draw a line down from the angle opposite the hypotenuse to the centre. It states that for any triangle inscribed inside the circle with all points touching the circumference and the hypotenuse as a diameter, then the angle opposite the hypotenuse will be right-angled.Ĭircle Theorem 1: The angle in a semicircle is 90 °, Ben Cairns, Vaia Proof of theorem 1: The angle in a semicircle is 90° This circle theorem is illustrated below. Theorem 1: The angle in a semicircle is 90° We will then look at how to apply these theorems. Let's look at each of the circle theorems, and then their proofs.
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