Triangle and two circles theorem

[Delong]

Can someone help me understand why this is the case? I tried forming a cartesian equation for the two circles but there were too many variables that it would be too messy to compute. Otherwise I am rather stuck on how to do it. I would appreciate it if someone can explain how to prove this theorem in accessible terms. Thanks.

In a triangle the inscribed circle touches the circle that passes through the midpoints of the sides.

 

[Studiot]

I have no idea what the 'two circles theorem' is but you seem to be discussing the incircle for a triangle.

The inscribed circle for any polygon is the circle to which each side is tangent.

To see why the tangent point is the mid point of each side look at my first sketch for some preliminary results.

From any point P outside any circle, centre O, two tangents may be drawn meeting the circle at M and N.
Since OM and ON are radii and NP and MP are tangents [tex]\hat{ONP}[/tex] and [tex]\hat{OMP}[/tex] are right angles.
Since OP is common to both triangles OMP and ONP they are congruent.

Therefore [tex]\hat{NPO}[/tex] = [tex]\hat{MPO}[/tex]
Therefore OP bisects [tex]\hat{NPM}[/tex] and the two tangents
Therefore OP bisects any line such as MN, crossing PM and PN produced

Taking this result into my second diagram you can see that the line from each vertex of triangle ABC to the centre of the incircle produced bisects the vertex angle and the opposite side.
Further these lines meet the opposite sides at right angles, since OF, OG, OH are radii and the sides are tangents.
Further these lines meet at the centre of the incircle.

go well

 

[AlephZero]

See http://mathworld.wolfram.com/TangentCircles.html

Specifically, the diagram after equation 16.

 

[Delong]

So yes I will read these very soon. Thanks!

 

[blue_raver22]

The triangle and two circles theorem, also known as the Feuerbach's theorem, states that the inscribed circle of a triangle is tangent to the circle that passes through the midpoints of the sides. This is a well-known and fundamental theorem in geometry, and its proof requires some basic concepts and techniques from geometry.

To understand why this is the case, let's consider the properties of the two circles involved. The inscribed circle is the largest circle that can be drawn inside the triangle, touching all three sides. The circle passing through the midpoints, also known as the nine-point circle, is the smallest circle that can be drawn outside the triangle, touching all three sides at their midpoints.

Now, let's imagine the inscribed circle and the nine-point circle as two gears that are interlocked, with the inscribed circle rotating inside the nine-point circle. As the inscribed circle rotates, it will always remain tangent to the nine-point circle, because they are both touching the sides of the triangle at the same points. This is similar to how two gears remain in contact as they rotate together.

To prove this theorem in mathematical terms, we can use the concept of power of a point. The power of a point is a measure of the distance of a point from a circle, and it is equal to the square of the length of the tangent drawn from that point to the circle. In this case, the power of the point where the inscribed circle and the nine-point circle touch is the same, as they are both tangent to the same sides of the triangle. This means that the two circles are actually tangent to each other at this point, which proves the theorem.

I hope this explanation helps you understand why the inscribed circle of a triangle is tangent to the circle passing through the midpoints of the sides. It is a beautiful and elegant result in geometry, and its proof highlights the interconnectedness and symmetry of geometric shapes. Keep exploring and discovering the wonders of mathematics and science!