Circles in a triangle
Kapitoly: Triangle, The height of a triangle, The weight of a triangle, Circles in a triangle, Right triangle, How to draw a triangle, Area of the triangle, The Pythagorean Theorem
For each triangle, we can draw a circle that is circumscribed or inscribed. The inscribed circle goes through all the points of the triangle and the inscribed circle touches all three sides of the triangle.
The inscribed circle of a triangle
A circle inscribed is a circle that passes through all vertices of a triangle. This circle always exists and is also the only one. Thus we can say that there is a circle passing through each of the three points that are not on the same line.
In order to draw the circle circumscribed by a triangle, we will need to know the concept of a side axis. The axis of a side c is a line that is perpendicular to the side c and also passes through the center of the side c, the point Sc. How to find the center of the side is described in the construction of the weights. Once we have the center of the side, we can run a perpendicular line through this point to create the side axis.
Each triangle has three sides, so it also has three side axes. These side axes intersect at one point, usually denoted by S. This point is then the centre of the circle being traced. So we proceed as follows. At the beginning, we are given a triangle and we want to draw a circle circumscribed by this triangle:
First we draw the axes of all three sides. That is, we find the centers of all three sides and then we run a perpendicular line through each center.
Now all that's left to do is draw the circle. The circle drawn goes through all the vertices, so the radius of such a circle is easy to find. It is the distance from the center S to any vertex.
A inscribed circle
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A circle is inscribed by a triangle if it touches all three sides, that is, it has just one point in common with each side. To find the center of an inscribed circle, we need to know the concept of the axis of an angle, which is already described and explained in the article on angles. The centre of the inscribed circle is then located at the intersection of all the axes of the angles. So at the beginning we have an ordinary triangle. Draw the axes of the angles:
The intersection S marks the center of the inscribed circle. We still need to find the radius. We will run a perpendicular line to any side that will just pass through the point S.
The length of the line SPc is equal to the length of the radius of the inscribed circle. Now all that's left to do is draw the circle:
