ABC is a triangle with a right angle at A. M and N are points on BC such that AM is the altitude, and AN is the angle bisector of ∠BAC. If CN/NB=21, what is CM/MB?

Note: Segment lengths BA and AB assumed to be equivalent in the following proof.

BN/sin(BAN)=BA/sin(BNA) (sine rule)
=> BA/BN=sin(BNA)/sin(45)

AC/sin(CNA)=NC/sin(45) (sine rule)
=> AC/NC=sin(CNA)/sin(45)

Since sin(BNA)=sin(180-BNA)=sin(CNA)

BA/BN=AC/NC, or
CN/NB =AC/AB=tan(B)

By metric relations
AM^2=BM*MC

By similar triangles,
AM/AB=MC/AC
=>AM=AB*MC/AC

Substitute in above
AB²*MC²/AC² = BM*MC
=> BM/MC = (AB/AC)^2
CM/MB=(AC/AB)^2=tan²(B)

To find the ratio CM/MB, we will use the Angle Bisector Theorem.

The Angle Bisector Theorem states that in a triangle, if an angle is bisected, then the ratio of the lengths of the sides that form the angle is equal to the ratio of the lengths of the segments of the opposite side.

In triangle ABC, let CN = x and NB = 21x (given that CN/NB = 21).

Since AM is the altitude, it is perpendicular to BC. This means that triangle AMN is similar to triangle ABC by the Angle-Angle Similarity Theorem, as they share an angle at A and have a right angle at M. Therefore, we can set up the following proportion:

AM/AN = NM/AB

Substituting the given values, AM/AN = x / (21x + x) = x / 22x.

By the Angle Bisector Theorem, we know that AN/BN = AM/BM.

Substituting the known values, x / 21x = AM/BM.

Simplifying the equation, we have 1/21 = AM/BM.

Since AM is the altitude, it divides BC into CM and MB. So, CM + MB = BC.

Since triangle ABC is a right triangle with a right angle at A, we can use the Pythagorean Theorem to relate the sides:

AB^2 = AC^2 + BC^2.

Since angle BAC is a right angle, we have:

AB^2 = AC^2 + BC^2 = AM^2 + CM^2 + CN^2.

AM is the altitude, so AM^2 + CM^2 = AC^2.

Substituting the known values from our proportion, we have:

(22x)^2 = AC^2 + CN^2.

441x^2 = AC^2 + x^2.

Simplifying the equation, we have:

440x^2 = AC^2.

Now, let's denote BC as 22x to work with a single variable.

From the Pythagorean Theorem, we know that AB^2 = AC^2 + BC^2.

Substituting the known values, we have:

AB^2 = AC^2 + (22x)^2.

Since angle BAC is a right angle, AB^2 = BC^2 + AC^2.

Substituting the known values, we have:

(22x)^2 = (22x)^2 + AC^2.

Simplifying the equation, we have:

0 = AC^2.

Since AC must be positive, we conclude that AC = 0.

This means that triangle ABC is degenerate, with side BC collapsing onto side AM.

As a result, CM = AM and MB = BM.

Therefore, CM/MB = AM/BM = 1/1 = 1.