Unique Circle Equal Angles
Triangles ABC and ABD share the base AB.
Prove that a unique circle passes through all four points A, B, C, and D, iff (if and only if) α = β.
First of all we shall show that a unique circle exists that will pass through three vertices of any given triangle. Consider a general triangle ABC. We know that all points on the perpendicular bisector of AB will be equidistant from A and B. Similarly all points on the perpendicular bisector of BC will be equidistant from B and C. Hence the intersection of these two perpendicular bisectors, O, will be equidistant from A, B, and C, and will mark the centre of the circle passing through all three vertices, and clearly two lines intersect at a unique point.
Now we shall prove that in a given circle all angles on the same circumference from a common chord will be equal. We shall achieve this by showing that the centre angle is twice the angle at the circumference. Consider the following diagram.
As OA, OB, and OC are all radii, triangles OAC and OBC are isosceles. Hence 2α + x = 180 and 2β + y = 180. Also x + y + z = 360.
Therefore x + y + z = 2α + x + 2β + y, so z = 2α + 2β = 2(α + β). That is, the angle at the centre if twice the angle at the circumference.
As the point on the circumference, C, was arbitrarily chosen it should be clear that all angles on the circumference will be half the centre angle, and thus equal.
Now we are able to show in the original diagram that if α = β then a unique circle must pass through all four points, A, B, C, and D.
Given triangle ABC we know that a unique circle passes through each of the vertices. Now suppose that the circle does not pass through D, but passes through some other point on the line through BD: E1 or E2.
We have just shown that all angles on the circumference are equal, so whichever point the circle passes through on the line through BD will also produce an angle α. It should be clear that there can be only one point on this line for which the angle will be α. Hence if the angle at D is equal to α we know that the unique circle which passes through A, B, and C, must also pass through D.
Prove that the centre of the circle will be found on the edge of the triangle iff it is a right angle triangle.