内分点計算機

The Section Formula finds the point that divides a line segment in a given ratio. It generalizes the midpoint formula — the midpoint is the special case where the ratio is 1:1. The section formula has two versions: internal division (the point lies between the two endpoints) and external division (the point lies outside the segment, on its extension). This tutorial covers both, derives them from the principle of similar triangles, and walks through worked examples for each.

The internal division formula

Given two points P₁ = (x₁, y₁) and P₂ = (x₂, y₂), the point P that divides the segment P₁P₂ internally in the ratio m:n is:

P = ( (mx₂ + nx₁) / (m + n),   (my₂ + ny₁) / (m + n) )

The point P is BETWEEN P₁ and P₂. The ratio m:n means P is m units away from P₁ for every n units away from P₂. (So if m > n, P is closer to P₂; if m < n, P is closer to P₁.)

Special case — the midpoint

Setting m = n = 1 gives:

P = ( (1·x₂ + 1·x₁) / 2, (1·y₂ + 1·y₁) / 2 ) = ( (x₁ + x₂) / 2, (y₁ + y₂) / 2 )

That's the midpoint formula. The midpoint divides the segment in ratio 1:1 — equidistant from both endpoints.

Where the formula comes from

The internal division formula follows from similar triangles. Imagine the segment P₁P₂ on a coordinate plane. Drop perpendiculars from P₁, P, and P₂ to the x-axis. The three resulting horizontal positions are x₁, x_P, and x₂.

By similar triangles, the ratio of horizontal positions matches the ratio in which P divides the segment:

(x_P − x₁) / (x₂ − x_P) = m / n

Cross-multiply: n(x_P − x₁) = m(x₂ − x_P)
n · x_P − n · x₁ = m · x₂ − m · x_P
x_P (m + n) = m · x₂ + n · x₁
x_P = (m · x₂ + n · x₁) / (m + n)

Same logic for y_P. Combining gives the section formula.

External division

If P lies on the LINE through P₁ and P₂ but OUTSIDE the segment (beyond one of the endpoints), we say P divides the segment externally in ratio m:n.

The formula is similar but with a sign flip:

P_ext = ( (mx₂ − nx₁) / (m − n),   (my₂ − ny₁) / (m − n) )

Same form, but with subtraction instead of addition in both the numerator and the denominator.

Equivalent trick: external division in ratio m:n is internal division in ratio m:(−n), or equivalently in ratio (−m):n. Our calculator handles both — enter n as negative for external division.

Worked example 1 — internal division

Find the point that divides the segment from P₁ = (1, 2) to P₂ = (7, 8) in the ratio 2:1 (internal).

m = 2, n = 1, m + n = 3.

x_P = (2 · 7 + 1 · 1) / 3 = (14 + 1) / 3 = 15/3 = 5
y_P = (2 · 8 + 1 · 2) / 3 = (16 + 2) / 3 = 18/3 = 6

P = (5, 6). Verify: distance from (1,2) to (5,6) is √(16+16) = √32 ≈ 5.66. Distance from (5,6) to (7,8) is √(4+4) = √8 ≈ 2.83. The ratio is 5.66 : 2.83 ≈ 2 : 1. ✓

Worked example 2 — midpoint via section formula

Find the midpoint of P₁ = (2, −3) and P₂ = (8, 5). Use the section formula with m = n = 1:

x_M = (1 · 8 + 1 · 2) / 2 = 10/2 = 5
y_M = (1 · 5 + 1 · (−3)) / 2 = 2/2 = 1

M = (5, 1). Same answer the standard midpoint formula gives.

Worked example 3 — external division

Find the point that divides P₁ = (1, 2) and P₂ = (4, 5) externally in ratio 3:2.

x_P = (3 · 4 − 2 · 1) / (3 − 2) = (12 − 2) / 1 = 10
y_P = (3 · 5 − 2 · 2) / (3 − 2) = (15 − 4) / 1 = 11

P = (10, 11). This point lies on the line through P₁ and P₂, beyond P₂ (extending the segment).

3D extension

Just like the midpoint formula, the section formula extends to three dimensions by adding a z-coordinate term:

P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n))

Every component (x, y, z) divides in the same ratio.

Centroid of a triangle — a section-formula application

The centroid (intersection of the three medians) of a triangle with vertices A, B, C is at:

centroid = ((x_A + x_B + x_C)/3, (y_A + y_B + y_C)/3)

This is a 2:1 section of every median. The centroid divides each median (from a vertex to the opposite midpoint) in the ratio 2:1 from the vertex. Applying the section formula to (vertex) : (midpoint of opposite side) with ratio 2:1 gives the centroid above.

The 1/3 averaging form is the simplification you get from working out the section formula for this special case.

Real-world applications

• Surveying and mapping. Locating a point on a line at a given fraction of the way between two known points.
• Computer graphics. Animation interpolation: position at time t along a path P₁ → P₂ is the t : (1−t) section, often written as P(t) = (1−t)P₁ + tP₂. Same idea as the section formula.
• Physics — center of mass. The center of mass of two point masses m₁ at P₁ and m₂ at P₂ is the section of P₁P₂ in ratio m₂ : m₁ (heavier point pulls the COM closer to it).
• Architecture. Dividing a beam, column, or facade at proportional positions for aesthetic or structural purposes (the golden ratio φ ≈ 1.618 is a famous example).

Common mistakes

• Reversing m and n in the formula. The internal formula has m multiplying x₂ and n multiplying x₁ — that is, m corresponds to the FAR point. Reversing them gives a different point.
• Confusing internal with external. Internal division puts P between P₁ and P₂. External division puts P outside. Check whether your problem says "internally" or "externally" or implies which by context.
• Forgetting to simplify the ratio. The point dividing in ratio 4:6 is the same as the point dividing in ratio 2:3. Simplifying gives the same answer with smaller numbers.
• Using the formula on a non-collinear point. The section formula always produces a point on the line through P₁ and P₂ — if you want a point not on that line, the section formula is not the right tool.