# Prove that the geometric mean is always the same

• I
• Trysse
In summary, the conversation discusses the Intersecting Chords Theorem which states that the product of the distances from a fixed point ##P## to two points on a circle intersected by two lines through ##P## is always the same, regardless of the orientation of the lines. The theorem is proven by demonstrating that the triangles formed by the points on the circle and the fixed point are similar. This result can be observed in the GeoGebra model provided and is not limited to a specific circle or point.

#### Trysse

Given are a fixed point ##P## and a fixed circle ##c## with the radius ##r##. Point ##P## can be anywhere inside or outside the circle. I now draw two arbitrary lines ##l_1## and ##l_2## through the point ##P## in such a way, that both lines intersect with the circle ##c## in two distinct points. I name the points ##C_1## and ##C_2## for the line ##l_1##, respectively ##C_3## and ##C_4## for the second line ##l_2##. This gives me two sets of distances. For ##l_1## these are the distances ##PC_1## and ##PC_2##. For ##l_2## these are the distances ##PC_3## and ##PC_4##

For reference see the GeoGebra model: https://www.geogebra.org/classic/xvgereug

According to the GeoGebra model, the product of the two distances always has the same value for a given point ##P## and a given circle ##c## regardless of the orientation of the two lines.

$$PC_1 \cdot PC_2 = PC_3 \cdot PC_4$$

If I draw the line ##l_1## in such a way, that the distances ##PC_1=PC_2## this length is the square root of any two distances ##PC_3## and ##PC_4##. I.e.,

$$PC_1=PC_2 \Rightarrow PC_1=\sqrt {PC_3 \cdot PC_4}$$

So I have two questions:
1)
How can I prove, that the geometric mean (or the product) of the two distances ##PC_1## and ##PC_2## is always the same for any line through the point ##P## that intersects the circle ##c## in two points?

2)
I was thinking, that the value of the geometric mean of these two distances is somehow the geometric mean of all distances the point ##P## has to all points that lie on the circle ##c##. However, I found this post on StackExchange that argues otherwise. My intuitive approach was to calculate the following:

$$\sqrt[4] {PC_1 \cdot PC_2 \cdot PC_3 \cdot PC_4}$$

As I add ever more lines ##l_n## so I get more distances ##PC_{2n-1}## and ##PC_{2n}## that I can add in pairs to calculate the geometric mean, the result will always be the same. What is wrong with my approach? For the StackExcnage post, see here:

https://math.stackexchange.com/ques...stances-from-point-to-every-point-on-a-circle

In the Geogebra model, you can set the point ##P## with drag-and-drop and then change the lines' orientation by moving the pink points.

This is not homework.

Last edited:
What you are trying to prove is the "Intersecting Chords Theorem". There's a nice, short, simple proof here.
The result follows from demonstrating that triangles ##PC_1C_4## and ##PC_3C_2## are similar, because angles on the circle at the circumference standing on the same chord are equal.

BvU, hutchphd, jim mcnamara and 1 other person
@andrewkirk thanks. That is what I was looking for.. I would have never found this theorem on Google. I was always searching for "geometric mean, & circle & proof" but never for "chords".... Thanks for pointing me in the right direction.