Triangle Vertices - From Three Midpoints
Use this triangle vertices calculator to reconstruct the three corners A, B, and C from the midpoints D, E, and F of each side using the midpoint formula.
Triangle Vertices
Results
What Is the Triangle Vertices Calculator?
The triangle vertices calculator reconstructs the three corner coordinates A, B, and C of a triangle from the three midpoints D, E, and F of its sides, using the midpoint formula rearranged to solve for each missing endpoint. Type the six midpoint coordinates and the page returns the matching vertex pair for each corner.
- • Coordinate geometry homework: Verify a textbook problem where the three midpoints are given and the question asks for the three corner coordinates.
- • Survey sketch recovery: Convert a triangle whose three side midpoints were marked on a paper plan into the three corner coordinates needed for an area report or a CAD import.
- • Median and centroid analysis: Build a triangle from its side midpoints to study the medians and the centroid without first having the vertices.
A triangle in the plane is fixed by three non-collinear points, but it is also fixed by the three midpoints of its sides, because the midpoint formula is a linear system that can be inverted.
For the forward direction, from two vertices to a single midpoint, the Midpoint Calculator in the same Math & Conversion cluster applies the same averaging step that this page inverts.
How the Triangle Vertices Calculator Works
The calculator takes the six midpoint coordinates, applies the vertex-from-midpoint formula to each corner, and returns the three (x, y) vertex pairs as primary outputs. The same arithmetic is shown as a worked example so the result can be checked by hand.
- x1, y1: Coordinates of midpoint D, the midpoint of side BC opposite vertex A.
- x2, y2: Coordinates of midpoint E, the midpoint of side AC opposite vertex B.
- x3, y3: Coordinates of midpoint F, the midpoint of side AB opposite vertex C.
- Ax, Ay: Vertex A, computed as (x2 + x3 - x1, y2 + y3 - y1), the sum of the midpoints on the two sides that meet at A (E and F) minus the midpoint on the opposite side (D).
- Bx, By: Vertex B, computed as (x1 + x3 - x2, y1 + y3 - y2), the sum of the midpoints on the two sides that meet at B (D and F) minus the midpoint on the opposite side (E).
- Cx, Cy: Vertex C, computed as (x1 + x2 - x3, y1 + y2 - y3), the sum of the midpoints on the two sides that meet at C (D and E) minus the midpoint on the opposite side (F).
Each vertex is the sum of the two midpoints on the sides that meet at that vertex minus the third midpoint, so each coordinate is a single addition and subtraction.
Worked example
Midpoints D = (2, 3), E = (4, 3), F = (3, 1)
A = (4 + 3 - 2, 3 + 1 - 3) = (5, 1). B = (2 + 3 - 4, 3 + 1 - 3) = (1, 1). C = (2 + 4 - 3, 3 + 3 - 1) = (3, 5).
A = (5, 1), B = (1, 1), C = (3, 5).
Re-running the midpoint formula on each pair of vertices confirms the same three midpoints: midpoint of B and C is D, midpoint of A and C is E, midpoint of A and B is F.
According to Wikipedia, the centroid of a triangle is the arithmetic mean of its three vertices and is also the common point of the three medians, each of which connects a vertex to the midpoint of the opposite side.
According to Wolfram MathWorld, the midpoint of a segment with endpoints (x_a, y_a) and (x_b, y_b) is the arithmetic mean M = ((x_a + x_b)/2, (y_a + y_b)/2), which rearranges to x_a = 2M - x_b for the missing endpoint.
Once the three vertices are in hand, the Area Triangle Coordinates Calculator in the same Math & Conversion cluster reads the shoelace area in the same length unit as the input midpoints.
Key Concepts Explained
Four ideas decide whether the vertex-from-midpoint formula is the right tool and whether the result is a real triangle.
Midpoint of a side
The midpoint of a segment is the point equidistant from the two endpoints, and in coordinates it is the arithmetic mean of the two endpoint coordinates. The triangle vertices calculator starts from the three side midpoints and walks the averaging step backwards.
Midpoint theorem and medians
The segment that connects a vertex to the midpoint of the opposite side is called a median, and the three medians all pass through the centroid. Knowing the three side midpoints is enough to recover the three medians and the vertices.
Reverse averaging
Because the midpoint formula is linear, it can be inverted: given the midpoint and one endpoint, the other endpoint is 2M - endpoint. The vertex formula uses that fact twice per vertex.
Centroid as a sanity check
The centroid of the recovered triangle is the arithmetic mean of the three output coordinates, and the centroid of the midpoint triangle is the arithmetic mean of the six input coordinates. The two should match, the easiest way to confirm a hand result.
When the three midpoints coincide, the formula returns the same point three times, the correct answer for a degenerate triangle of zero area.
The arithmetic mean of the three recovered vertex pairs is the centroid of the triangle, which the Centroid in the same category reports directly from the three vertex coordinates.
How to Use This Calculator
Type the six midpoint coordinates in the same length unit, then read the three vertex pairs in the results panel.
- 1 Enter midpoint D: Type the x and y coordinates of the midpoint of the side opposite vertex A, in the same length unit for all six values.
- 2 Enter midpoint E: Type the x and y coordinates of the midpoint of the side opposite vertex B. The order matters: D is opposite A, E is opposite B, F is opposite C.
- 3 Enter midpoint F: Type the x and y coordinates of the midpoint of the side opposite vertex C. The results panel updates the three vertex pairs in real time.
- 4 Read vertex A: The first row of the results panel shows vertex A, computed as (x2 + x3 - x1, y2 + y3 - y1), the corner opposite the midpoint D.
- 5 Read vertices B and C: The next two rows give vertices B and C using the same midpoint formula, the unique triangle that matches the three midpoints.
- 6 Sanity-check the result: Take the recovered vertices, run them through the midpoint formula, and confirm the same three midpoints come back. A match confirms the calculation.
For midpoints D = (2, 3), E = (4, 3), F = (3, 1), the results panel shows A = (5, 1), B = (1, 1), C = (3, 5). The midpoint of B and C is D, the midpoint of A and C is E, and the midpoint of A and B is F, matching the input midpoints exactly.
After reading the three vertex pairs, the Classifying Triangles Calculator in the same Math & Conversion cluster takes the same three pairs and reports whether the triangle is acute, right, or obtuse.
Benefits of Using This Calculator
The vertex-from-midpoint formula is one of the small number of coordinate-geometry results that stays the same shape no matter how the triangle is rotated or where it sits on the grid.
- • One pass from midpoints to vertices: The same three midpoints feed all three vertex formulas, so six input numbers become six output numbers without an intermediate side-length or angle calculation.
- • Works in any orientation: The formula is purely arithmetic, so a rotated, reflected, or offset triangle produces the same algebraic result, only translated to the new position.
- • Centroid and area follow-ups: The recovered vertices feed the centroid formula and the shoelace area formula in the same cluster, so the next step is a single click on a related page.
- • Negative and decimal coordinates work: Midpoints in the third quadrant and midpoints with one or five decimals all run through the same expression without a special case.
- • Hand-checkable on paper: Each vertex is the sum of two midpoints minus the third, the same expression a student would write on an exam.
When the three midpoints are given as a list of points on a grid, the Coordinate Plane Calculator in the same category plots the same three points on a labeled plane, which makes the order of D, E, and F obvious before the vertex formula is applied.
Factors That Affect Your Results
The formula is short, but a few input choices decide whether the calculator returns a triangle that actually matches the midpoints.
Order of the three midpoints
D must be opposite A, E opposite B, and F opposite C. Swapping D and E swaps vertices A and B in the result. The vertex labels move, but the midpoints are still right.
Length unit for all six coordinates
Mixing meters and feet silently produces a triangle in the wrong unit, so use one unit and convert the result later if needed.
Collinear midpoint triple
When the three midpoints lie on a single line, the recovered vertices are also collinear and the triangle has zero area. The formula does not raise an error, but the answer is degenerate.
Coincident midpoints
If two of the three midpoints are equal, two of the three vertices collapse to the third midpoint. The result is a degenerate triangle of zero area, with no arithmetic error.
Decimal precision of the input
The page keeps full double precision and only rounds the display to four decimal places, so very small differences between nearby midpoints still resolve to distinct vertices.
- • The calculator assumes the three midpoints come from a single triangle. If the three midpoints belong to two different triangles, the formula still returns three points, but they are not the corners of either triangle.
- • The output is a vertex pair in the same length unit as the input. The page does not convert between meters, feet, or any other unit.
According to Omni Calculator, the three vertices of a triangle can be recovered from the three side midpoints using a single arithmetic step per vertex: each corner is the sum of the two midpoints on the sides that meet at that corner, minus the midpoint of the opposite side.
When the recovered triangle is a right triangle, the Right Triangle Calculator in the same Math & Conversion cluster reads the same three vertices and reports the two legs, the hypotenuse, and the right angle.
Frequently Asked Questions
Q: How do you find the vertices of a triangle from the midpoints of its sides?
A: Label the three midpoints D, E, F of the sides opposite A, B, C. Vertex A is (x2 + x3 - x1, y2 + y3 - y1), B is (x1 + x3 - x2, y1 + y3 - y2), C is (x1 + x2 - x3, y1 + y2 - y3).
Q: What is the formula for triangle vertices using midpoints?
A: Each vertex coordinate is the sum of the two midpoints on the sides that meet at that vertex minus the third midpoint, the midpoint formula inverted to solve for the missing endpoint.
Q: How many midpoints are needed to reconstruct a triangle?
A: Three midpoints are enough, and they must come from a single triangle. Two midpoints fix one segment but leave the third vertex free, and four midpoints would over-determine the triangle.
Q: Can the vertices be found if the midpoints are collinear?
A: The arithmetic still works, but the three recovered vertices are collinear too, so the result is a degenerate triangle with zero area.
Q: What happens if two of the three midpoints coincide?
A: Two of the three vertices collapse to the third midpoint, so the result is a triangle of zero area with one side of length zero.
Q: Do the three midpoints always form a triangle similar to the original?
A: Yes, the medial triangle is similar to the original with a scale factor of 1/2 and the same angles. Multiplying the medial side lengths by 2 is an independent way to recover the original side lengths.