Inscribed circle in the triangle

Click For Summary
SUMMARY

The discussion centers on the geometric properties of triangle ABC, specifically regarding the inscribed circle's center, point I. It establishes that the heights of triangle PQD intersect at point I, the incenter of triangle ABC. The participants explore the relationships between points D, P, and Q, and their connections to the angle bisectors and perpendicular bisectors within the triangle. A diagram is suggested to aid in visualizing these relationships, confirming the validity of the geometric statement presented.

PREREQUISITES
  • Understanding of triangle properties, specifically incenter and orthocenter concepts.
  • Familiarity with angle bisectors and perpendicular bisectors in geometry.
  • Knowledge of basic geometric constructions and their implications.
  • Ability to interpret geometric diagrams and proofs.
NEXT STEPS
  • Study the properties of triangle centers, focusing on incenter and orthocenter relationships.
  • Learn about the construction and significance of angle bisectors in triangles.
  • Explore geometric proof techniques, particularly those involving perpendicular bisectors.
  • Investigate the use of geometric diagrams to enhance understanding of complex relationships in triangles.
USEFUL FOR

Mathematicians, geometry enthusiasts, and students studying triangle properties and geometric proofs will benefit from this discussion.

Mathick
Messages
23
Reaction score
0
In the triangle a point $$I$$ is a centre of inscribed circle. A line $$AI$$ meets a segment $$BC$$ in a point $$D$$. A bisector of $$AD$$ meets lines $$BI$$ and $$CI$$ respectively in a points $$P$$ and $$Q$$. Prove that heights of triangle $$PQD$$ meet in the point $$I$$.

I've tried to show that sides of triangle $$PQD$$ are parallel to sides of triangle $$ABC$$ but it didn't work out. That's why I ask you for help.
 
Mathematics news on Phys.org
Can you provide a diagram?
 
I am convinced the OP has posted a true statement. I can't prove it, but hope someone here can. Here's a diagram.

1hug6x.png


Give a triangle ABC, let I be the incenter of the triangle (the intersection of the angle bisectors at A, B and C). Let D be the intersection of AI with BC and line L the perpendicular bisector of AD ($H_D$ is the midpoint of AD). Let P be the intersection of CI with L and Q the intersection of BI with L. Then the orthocenter of PDQ is I.

Clearly, if $H_Q$ is the intersection of BI with DP, it is sufficient to show angle $BH_QD$ is a right angle.
 

Similar threads

High School In a triangle, is the angle between the points of contact of the inscribed circle with the sides 120 degrees?
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Geometry problem of interest with a 3-4-5 triangle
  • · Replies 59 ·
2
Replies
59
Views
80K
MHB Prove Triangle Inequality: AB/MZ + AC/ME + BC/MD ≥ 2t/r
  • · Replies 1 ·
Replies
1
Views
2K
MHB ASVAB circle and inscribed rectangle area problem
  • · Replies 6 ·
Replies
6
Views
2K
High School Inscribing a circle in an Oblique Square (Drafting)
  • · Replies 9 ·
Replies
9
Views
2K
High School A triangle problem: Prove that |AC|+|AB|=|PR|+|PQ| given some constraints
  • · Replies 13 ·
Replies
13
Views
3K
MHB Difference of angles in a triangle
  • · Replies 4 ·
Replies
4
Views
2K
MHB AO+BO+CO≥6r where r is the radius of the inscribed circle
  • · Replies 2 ·
Replies
2
Views
1K
MHB Proving Equality of Angles in an Acute Triangle
  • · Replies 4 ·
Replies
4
Views
1K
MHB Is the Midpoint of HE the Center of the Inscribed Circle in Triangle HBC?
  • · Replies 1 ·
Replies
1
Views
2K