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Symbol Defintion
Example : GT means greater than
Q01. Ortho-center
Defintion
- Three heights of triangle are concurrent at a poin O which is ortho-center
- Draw a triangle ABC
- Draw height AD perpendicualar to BC
- Draw height BE perpendicualar to CA
- Draw height CF perpendicualar to AB
- Three height meets at one point O which is the orthocenter
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Q02. Prove that three heights meet at one point by construction
- Draw a large triangle ABC
- Draw AP perpendicular to BC
- Draw BQ perpendicular to CA
- Draw CR perpendicular to AB
- Hence three heights are concurrent at O
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Q03. Three feets form a triangle PQR which is called pedal triangle
- Triangle APQ is similar as ABC
- Angle CQP + angle AQP = 180 degrees (supplementary)
- Angle AQP + angle ABC = 180 degrees (concyclic)
- Hence angle CQP = angle ABC
- Angle C is in common for these two triangle
- Hence they are similar
- Similarly, triangle AQR is similar to triangle ABC
- Similarly, triangle AQR is similar to triangle ABC
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Q04. Pedal triangle : the feets of the heights form a triangle
Construction
- Draw a large triangle
- Draw AP perpendicular to BC
- Draw BQ perpendicular to CA
- Draw CR perpendicular to AB
- Hence three feets of the heights are P, Q, R
- Triangle PQR is tedal triangle of triangle ABC
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Q05. Ortho-center O of traingle ABC is also incenter of triangle PQR
Prove point O is in-center of triangle PQR
- Prove that BQ is bisector of angle PQR by measurement
- Prove that AP is besector of angle QPR by measurement
- Prove that CR is besector of angle PRQ by measurement
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Q06. Ortho-center : relation with circum-center
Construction
- Draw a large triangle
- Draw orthod-center O
- Draw circomcenter U
- Draw COF perpendicular to AB
- Draw UR Perpendicular to AB
- Proof : By measurement
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Q07. Angles of pedal triangle PQR and angles of triangle ABC
Angles of triangle PQR and angles of triangle ABC
- Prove that Angle PQR = 180 - 2*B by measurement
- Prove that Angle QRP = 180 - 2*C by measurement
- Prove that Angle RPQ = 180 - 2*A by measurement
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Q08. New
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Q09. New
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Q10. References
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