The relationship between triangles and circles

What is the relationship between triangles and circles? | Yahoo Answers IM Commentary. This task provides a good opportunity to use isosceles triangles and their properties to show an interesting and important result about triangles. Theorem I If a straight line touches a circle and from the point of contact a chord is drawn, the angles which this tangent makes with the chord. Start studying relationship between triangles and circles. Learn vocabulary, terms , and more with flashcards, games, and other study tools.

Now let's see what we can do to show this. Well, we have in our tool kit the notion of an inscribed angle, it's relation to a central angle that subtends the same arc.

So let's look at that. So let's say that this is an inscribed angle right here. Let's call this theta. Now let's say that that's the center of my circle right there. Then this angle right here would be a central angle.

Let me draw another triangle right here, another line right there. This is a central angle right here. This is a radius. This is the same radius -- actually this distance is the same.

But we've learned several videos ago that look, this angle, this inscribed angle, it subtends this arc up here. The central angle that subtends that same arc is going to be twice this angle. We proved that several videos ago. So this is going to be 2theta. It's the central angle subtending the same arc. Now, this triangle right here, this one right here, this is an isosceles triangle.

Illustrative Mathematics

I could rotate it and draw it like this. If I flipped it over it would look like that, that, and then the green side would be down like that. And both of these sides are of length r. This top angle is 2theta. So all I did is I took it and I rotated it around to draw it for you this way. This side is that side right there.

Since its two sides are equal, this is isosceles, so these to base angles must be the same. That and that must be the same, or if I were to draw it up here, that and that must be the exact same base angle.

Now let me see, I already used theta, maybe I'll use x for these angles. So this has to be x, and that has to be x. So what is x going to be equal to? Well, x plus x plus 2theta have to equal degrees.

Triangles and Circles - Pure - Geometry - Maths Reference with Worked Examples

They're all in the same triangle. So let me write that down. It appears that the triangle shown is an equilateral triangle. Equilateral triangles will be examined next to see if this holds true for a triangle that is constructed as an equilateral triangle.

To examine the five circles for a triangle using GSP, click here. When an equilateral triangle is constructed, the incircle and nine-point circle are the same circle. The two circles are shown as the red circle in the diagram below. It should be noticed that the nine-point circle, incircle and excircles are tangeant at midpoint of each side. As the triangle is changed, this relationship stays true. To examine the five circles for an equilateral triangle using GSP, click here. When an isosceles triangle is created with its five circles, the relationship that exists is different from the realtionship between an equilateral and its five circles.

The nine-point and incircle are tangeant to an excircle at the midpoint of the base. To examine the five circles for an isosceles triangle using GSP, click here. The last special triangle to be examined is a right triangle. The nine- point circle and three excircles are always tangeant at one point for a right triangle. A different result from when we looked at the equilateral and isosceles triangle, the tangeancy point does not have be tangeant to one of the sides of the triangle for any of three excircles. This can be seen in the diagram below. To examine the five circles for a right triangle using GSP, click here.