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.

The measures of angles of triangle MNP are

Step-by-step explanation:

The picture of the question in the attached figure

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

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.

substitute

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.

substitute

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.

substitute

find the magnitude of the trig term  

csc

(

x

)

−

1

by taking the absolute value of the coefficient.

1

find the lower bound of the range.

tap for more

y

=

−

2

find the upper bound of the range.

tap for more

y

=

0

the range is  

y

≤

−

2

or  

y

≥

0

.

(

−

∞

,

−

2

]

∪

[

0

,

∞

)

{

y

|

y

≤

−

2

,

y

≥

0

}

step-by-step explanation: