Thalet's circle
Kapitoly: Circles, Thalet's circle
Thalet's theorem says that all triangles whose center of the circle circumscribed lies on the hypotenuse of that triangle are right triangles.
Description of Thalet's circle
A certain Thales of Miletus long ago discovered an interesting property of the triangle. Let us draw a triangle and its circle circumscribed so that the centre of this circle will bisect the hypotenuse (longest side) of the triangle. Look at the following figure:
In the figure we have a circle k with its center at the point S. This circle is the hypotenuse of the triangle ABC, i.e. it passes through all the vertices of the triangle. The important property is that the hypotenuse passes through the center of the circle, passing through the point S.
Then, the interior angle ABC always has size $90^{\circ}$, it is a right angle.
No matter where we move the vertex B along the circle, we always get a right angle at that vertex. A few more examples:
Proof
We will deduce why a triangle in a Thalet circle is always right-angled. Let's modify the first figure a little:
It is true that the triangles SCB and ASB are isosceles because they always have two sides of equal length. For the triangle SCB, the sides are SC and SB, because their length is equal to the radius of the circle. For the other triangle, the sides are SA and SB.
Therefore, the magnitude of the angle BCS and the angle SBC are the same, shown in the figure by the letter β. The same for the second triangle, there it is indicated by the letter α.
We also know that the sum of the angles in the triangle must always give us $180^{\circ}$. Now we express the sum of the angles in the triangle ABC using the angles α and β. It must be true
$$ \alpha + \beta + \alpha + \beta = 180. $$
One angle α is there for the angle CAB, the other angle α for the angle ABS. Do the same for the angle β. We will modify the previous equation slightly more:
$$ 2\alpha+2\beta = 180 $$
Divide the equation by two:
$$ \alpha+\beta = 90 $$
And the proof is complete. We know that the sum of the angles α + β is equal to 90, and we know that the sum of the angles α and β gives us the interior angle ABC.