Question

Acute △ABC with angles α, β, and γ is inscribed in a circle. Tangents to the circle at points A, B, and C intersect in points M, N, and P. Find measures of angles of the △MNP.

Answer 1

Answer:

The measures of angles of triangle MNP are

$M=[\gamma+\alpha-\beta]$

$N=[\beta+\alpha-\gamma]$

$P=[\beta+\gamma-\alpha]$

Step-by-step explanation:

step 1

Find the measure of arcs AB, BC and AC

we know that

The inscribed angle is half that of the arc it comprises.

so

$\gamma=\frac{1}{2}[arc\ AB]$ ----> $arc\ AB=2\gamma$

$\alpha=\frac{1}{2}[arc\ BC]$ ----> $arc\ BC=2\alpha$

$\beta=\frac{1}{2}[arc\ AC]$ ----> $arc\ AC=2\beta$

step 2

Find the measure of angle M

we know that

The measurement of the outer angle is the semi-difference of the arcs it encompasses.

$M=\frac{1}{2}[arc\ AB+arc\ BC-arc\ AC]$

substitute

$M=\frac{1}{2}[2\gamma+2\alpha-2\beta]$

$M=[\gamma+\alpha-\beta]$

step 3    

Find the measure of angle N

we know that

The measurement of the outer angle is the semi-difference of the arcs it encompasses.

$N=\frac{1}{2}[arc\ AC+arc\ BC-arc\ AB]$

substitute

$N=\frac{1}{2}[2\beta+2\alpha-2\gamma]$

$N=[\beta+\alpha-\gamma]$

step 4    

Find the measure of angle P

we know that

The measurement of the outer angle is the semi-difference of the arcs it encompasses.

$P=\frac{1}{2}[arc\ AC+arc\ AB-arc\ BC]$

substitute

$P=\frac{1}{2}[2\beta+2\gamma-2\alpha]$

$P=[\beta+\gamma-\alpha]$