Apollonius's theorem is an elementary geometry theorem relating the length of a median of a triangle to the lengths of its sides. While most of the world refers to it as it is, in East Asia, the theorem is usually referred to as Pappus's theorem or midpoint theorem. It can be proved by Pythagorean theorem from the cosine rule as well as by vectors. The theorem is named after a Greek mathematician Apollonius of Perga.
For triangle with being the midpoint of , the following equation holds:
Prove that triangle with being the midpoint of satisfies Apollonius' theorem by using only the Pythagorean theorem.
Let be the foot of perpendicular from to .
It is clear that
Using the Pythagorean theorem, we get
From these, we can conclude that
Prove that triangle with being the midpoint of satisfies Apollonius's theorem by using only the cosine rule (law of cosines).
Using the cosine rule, we get
Adding these two gives
Prove that triangle with being the midpoint of satisfies Apollonius's theorem only by using elementary vector operations.
Let be the origin of a Cartesian coordinate system.
Defining and it's clear that and
Substitute into Stewart's theorem, , and you get
Also, substituting and into Apollonius's theorem, you get
We can also obtain the Carnot's theorem, which reads
for any point Look for the Centroid of a Triangle wiki for proof.
Triangle has side lengths and as shown. is the midpoint of satisfying
Let be the point of intersection between and the bisector of meets with at point and the bisector of meets with at point An extension of meets with at point
Given that the area of is times larger than that of for some rational numbers and find the value of
This problem is a part of <Grade 10 CSAT Mock Test> series.