WildTrig18: Medians, altitudes and vertex bisectors

We discuss inversion in a circle, and medians, altitudes and vertex bisectors of a triangle. These are solutions to the final two questions of Quiz 1 in WildTrig16.

Category:

Science & Technology

Tags:

• Rational
• trigonometry
• inversion
• medians
• altitudes
• bisectors
• wildberger
• mathematics

[grichard24]

When you solve a quadratic and get two solutions, how do you know which one to reject? Do you always have to solve two equations and look for a common value?

[njwildberger]

You don't always have to solve two equations--this is a pleasant but uncommon situation.

More typically you can estimate which of the two solutions is right depending on the problem.

A more systematic answer is that the Triangle Spread Rules give a procedure for choosing---in many cases. These very useful rules are described in WildTrig 58 and 59.

[benthurston27]

Is the only time the 3 green lines meet at one point from problem 7 when the triangle is equilateral? I wonder when you have 3 unequal sides if the little triangle the 3 lines make in the center has any special property? It looks sort of like it might be similar to the whole triangle in the example.

[JanSobies]

I hadn't realized the triple spread formula could be used with three concurrent lines so I used a more complicated method to solve problem 4.

On problem 6 I used Thales' circle theorem to build right triangles from the diameter to F and E and then repeatedly used the spread law to derive the result.

[enzoslr]

thank you!