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# Triangles and Internal bisectors | PRMO 2019 | Question 10 Try this beautiful problem from the PRMO, 2019 based on triangles and internal bisectors.

## Triangles and internal bisectors - PRMO 2019

Let ABC be a triangle and let D be its circumcircle, The internal bisectors of angles A,B and C intersect D at $A_1,B_1 and C_1$ the internal bisectors of $A_1,B_1,C_1$ of the triangle $A_1B_1C_1$ intersect D at $A_2,B_2,C_2$. If the smallest angle of triangle ABC is 40 find the magnitude of the smallest angle of triangle $A_2B_2C_2$ in degrees.

• is 107
• is 55
• is 840
• cannot be determined from the given information

### Key Concepts

Lines

Algebra

Angles

PRMO, 2019, Question 10

Geometry Vol I to IV by Hall and Stevens

## Try with Hints

First hint

angle $A_1B_1C_1=90 - \frac{ABC}{2}$ angle $A_1C_1B_1=90-\frac{ACB}{2}$

Second Hint

angle $B_1A_1C_1$=90-$\frac{BAC}{2}$

Final Step

then angle $A_2B_2C_2=90-\frac{90-\frac{ABC}{2}}{2}$=45+$\frac{ABC}{4}$=55.

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Try this beautiful problem from the PRMO, 2019 based on triangles and internal bisectors.

## Triangles and internal bisectors - PRMO 2019

Let ABC be a triangle and let D be its circumcircle, The internal bisectors of angles A,B and C intersect D at $A_1,B_1 and C_1$ the internal bisectors of $A_1,B_1,C_1$ of the triangle $A_1B_1C_1$ intersect D at $A_2,B_2,C_2$. If the smallest angle of triangle ABC is 40 find the magnitude of the smallest angle of triangle $A_2B_2C_2$ in degrees.

• is 107
• is 55
• is 840
• cannot be determined from the given information

### Key Concepts

Lines

Algebra

Angles

PRMO, 2019, Question 10

Geometry Vol I to IV by Hall and Stevens

## Try with Hints

First hint

angle $A_1B_1C_1=90 - \frac{ABC}{2}$ angle $A_1C_1B_1=90-\frac{ACB}{2}$

Second Hint

angle $B_1A_1C_1$=90-$\frac{BAC}{2}$

Final Step

then angle $A_2B_2C_2=90-\frac{90-\frac{ABC}{2}}{2}$=45+$\frac{ABC}{4}$=55.

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