Triangle Vertices Calculator
Find triangle vertices A, B, C from the coordinates of midpoints D, E, F
Enter Midpoint Coordinates
Midpoint D (side BC)
Midpoint E (side AC)
Midpoint F (side AB)
Triangle Vertices Results
Formulas Used:
Vertex A: A = (x₁ + x₃ - x₂, y₁ + y₃ - y₂)
Vertex B: B = (x₁ + x₂ - x₃, y₁ + y₂ - y₃)
Vertex C: C = (x₂ + x₃ - x₁, y₂ + y₃ - y₁)
Example Calculation
Given Midpoints
Midpoint D (side BC): (2, 3)
Midpoint E (side AC): (4, 3)
Midpoint F (side AB): (3, 1)
Calculation Steps
Vertex A: A = (2 + 3 - 4, 3 + 1 - 3) = (1, 1)
Vertex B: B = (2 + 4 - 3, 3 + 3 - 1) = (3, 5)
Vertex C: C = (4 + 3 - 2, 3 + 1 - 3) = (5, 1)
Triangle Vertex-Midpoint Relationships
Midpoint D
Midpoint of side BC
D = ((B + C)/2)
Midpoint E
Midpoint of side AC
E = ((A + C)/2)
Midpoint F
Midpoint of side AB
F = ((A + B)/2)
Calculator Tips
The three midpoints uniquely determine the triangle vertices
Each vertex is calculated using all three midpoint coordinates
The centroid of vertices equals the centroid of midpoints
Works for any triangle in a coordinate plane
Understanding Triangle Vertices from Midpoints
What are Triangle Vertices?
The vertices of a triangle are the three corner points where the sides of the triangle meet. In coordinate geometry, each vertex is represented by an ordered pair (x, y) that specifies its position on the coordinate plane.
Midpoint Properties
- •A midpoint divides a line segment into two equal parts
- •Three midpoints uniquely determine the original triangle
- •The triangle formed by midpoints is similar to the original
- •The midpoint triangle has 1/4 the area of the original
Vertex Calculation Formulas
Given midpoints:
D(x₁, y₁) - midpoint of side BC
E(x₂, y₂) - midpoint of side AC
F(x₃, y₃) - midpoint of side AB
Vertex formulas:
A = (x₁ + x₃ - x₂, y₁ + y₃ - y₂)
B = (x₁ + x₂ - x₃, y₁ + y₂ - y₃)
C = (x₂ + x₃ - x₁, y₂ + y₃ - y₁)
Mathematical Principle: These formulas are derived from the midpoint formula and the fact that each vertex appears in exactly two of the three midpoint equations.