Question

Suppose ABC is a triangle on the unit sphere s(0,1) with Side lengths a, b, c. Suppose M = |ABI and N= |AC| such that IAMI = IMBI and |AN| = |NC|. Derive a formula for the length of |MN| in terms of the side lengths a, b, c.

Answer 1

To find the length of |MN|, we can use the Law of Cosines and the fact that IAMI = IMBI and |AN| = |NC|. Using the Law of Cosines, the formula for |MN| is |MN| = sqrt(4a² - 4accos(θ)).

Step-by-step explanation:

To find the length of |MN|, we can use the Law of Cosines. Let's denote the angle between AB and AC as θ. Since |AN| = |NC|, the angle θ is the same as the angle between AB and MN. Using the Law of Cosines, we have:

|MN|² = |AM|² + |AN|² - 2|AM||AN|cos(θ)

Since IAMI = IMBI and |AN| = |NC|, |AM| = |BM|. Let's denote the length of |AM| as x. Using the triangle ABI, we have:

x² = a² + a² - 2accos(θ)

Simplifying the equation, we get:

x² = 2a² - 2accos(θ)

Substituting |MN|² = |AM|² + |AN|² - 2|AM||AN|cos(θ), we have:

|MN|² = 2a² - 2accos(θ) + 2a² - 2accos(θ) - 2(a)(c)cos(θ)

|MN|² = 4a² - 4accos(θ)

Taking the square root of both sides, we get:

|MN| = sqrt(4a² - 4accos(θ))