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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

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

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

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

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

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

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

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