Concept Overview
A point of concurrency is a point where three special lines meet.
In triangles, different special lines meet at different important points.
Points of Concurrency explained through a friendly whiteboard conversation, examples, visuals, and practice for school students.
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Notes
Step 2
Formulae
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Examples
Step 4
Practice
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A point of concurrency is a point where three special lines meet.
In triangles, different special lines meet at different important points.
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Concurrent lines are lines that pass through the same point.
In a triangle, medians meet at the centroid. Altitudes meet at the orthocentre. Angle bisectors meet at the incentre. Perpendicular bisectors meet at the circumcentre.
Each point has a different meaning. The centroid is a balance point. The incentre is linked to a circle inside the triangle. The circumcentre is linked to a circle passing through the vertices.
The centroid divides each median in the ratio 2:1 from the vertex.
Meaning: A median joins a vertex to the midpoint of the opposite side.
Definition: The three medians of a triangle meet at the centroid.
The centroid lies twhirds of the way from each vertex along a median.
Meaning: An altitude is a perpendicular line from a vertex to the opposite side or its extension.
Definition: The three altitudes of a triangle meet at the orthocentre.
In an acute triangle, the orthocentre is inside the triangle. In a right triangle, it is at the righngle vertex.
Meaning: An angle bisector divides an angle into two equal angles.
Definition: The three angle bisectors of a triangle meet at the incentre.
The incentre is the centre of the circle that touches all three sides of the triangle.
Meaning: A perpendicular bisector cuts a side into two equal parts at 90 degrees.
Definition: The three perpendicular bisectors of a triangle meet at the circumcentre.
The circumcentre is the centre of the circle passing through all three vertices.
Easy
Question: What is the point where the three medians of a triangle meet?
Medium
Question: A centroid divides a median. The shorter part is 5 cm. Find the longer part.
Exam-level
Question: A triangle has angle bisectors meeting at point I. What is point I called and what circle is it linked to?
Mixing up incentre and circumcentre
Why it happens: Both are centres of circles.
Correct approach: Incentre is for the inside circle. Circumcentre is for the circle through vertices.
Using centroid ratio for all concurrency points
Why it happens: The 2:1 rule is memorised without context.
Correct approach: Use 2:1 only for centroid on a median.
Thinking every point is always inside the triangle
Why it happens: Only acute triangle diagrams are remembered.
Correct approach: Check the triangle type. Some points can lie outside.
What does concurrent mean?
Concurrent lines meet at one point.
Which point is the balance point of a triangle?
The centroid is the balance point.
Which centre is linked to the incircle?
The incentre is linked to the incircle.
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