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Endpoint Calculator

Recover the missing endpoint from one endpoint and the midpoint — in 2D, 3D, or any given ratio
Endpoint B given A(3, -4) and midpoint M(7, 1)
Endpoint given A(2, 5, -1) and midpoint (-3, 0, 4)
One end of a diameter is (-1, 5) and the centre is (4, -2) — find the other end
Point P(1, 6) divides AB in ratio 1:3 from A(-2, 3) — find B

The Endpoint Formula

The midpoint of a segment ABAB averages the coordinates:

M=(xA+xB2,  yA+yB2)M = \left(\frac{x_A + x_B}{2},\; \frac{y_A + y_B}{2}\right)

Finding a missing endpoint just solves that for BB. Multiply through by 22 and subtract:

xB=2xM−xA,yB=2yM−yAx_B = 2x_M - x_A, \qquad y_B = 2y_M - y_A

  B=(2xM−xA,  2yM−yA)  \boxed{\;B = (2x_M - x_A,\; 2y_M - y_A)\;}

Read as vectors it is obvious: B=A+2AM→B = A + 2\overrightarrow{AM} — walk from AA to MM, then take the same step again.

In 3D the pattern extends coordinate by coordinate: B=(2xM−xA,  2yM−yA,  2zM−zA)B = (2x_M - x_A,\; 2y_M - y_A,\; 2z_M - z_A).

What it assumes: the point you were handed really is the midpoint, at ratio 1:11:1. If it divides the segment in any other ratio, this formula is wrong and you need the section formula below.

Circles, 3D and Any Ratio

Diameter of a circle. The centre is the midpoint of any diameter, so given one end AA and centre CC, the other end is B=(2xC−xA,  2yC−yA)B = (2x_C - x_A,\; 2y_C - y_A) — the same formula.

The section formula. If PP divides ABAB in ratio m:nm:n measured from AA, then

P=(n xA+m xBm+n,  n yA+m yBm+n)P = \left(\frac{n\,x_A + m\,x_B}{m+n},\; \frac{n\,y_A + m\,y_B}{m+n}\right)

Setting m=n=1m = n = 1 recovers the midpoint. To find BB from AA and an interior point PP at ratio m:nm:n, rearrange to

B=A+m+nm (P−A)B = A + \frac{m+n}{m}\,(P - A)

Checking the answer takes one line and catches nearly every sign error: average your AA and BB and confirm you get MM back. Optionally verify ∣AM∣=∣MB∣|AM| = |MB| with the distance formula.

Common Mistakes to Avoid

  • Subtracting instead of doubling. B=M−AB = M - A is the vector from AA to MM, not the endpoint. The endpoint needs 2M−A2M - A.
  • Dividing by 2. The midpoint formula divides; the endpoint formula multiplies. Going backwards reverses the operation.
  • Sign slips with negatives. 2(1)−(−4)=2+4=62(1) - (-4) = 2 + 4 = 6. Write the substitution out before simplifying.
  • Swapping which point is the midpoint. Read the problem: the midpoint is the one described as "the middle", "the centre" or "the bisection point".
  • Assuming a given interior point is the midpoint. "Divides ABAB in a 1:31:3 ratio" is not a midpoint; use the section formula.
  • Dropping the third coordinate in a 3D problem — every coordinate gets the same treatment.

示例题目

Step 1: Use B=(2xM−xA,  2yM−yA)B = (2x_M - x_A,\; 2y_M - y_A)
Step 2: xB=2(7)−3=14−3=11x_B = 2(7) - 3 = 14 - 3 = 11
Step 3: yB=2(1)−(−4)=2+4=6y_B = 2(1) - (-4) = 2 + 4 = 6
Step 4: Check: midpoint of (3,−4)(3,-4) and (11,6)(11,6) is (3+112,−4+62)=(7,1)\left(\tfrac{3+11}{2}, \tfrac{-4+6}{2}\right) = (7, 1) ✓
Answer: B=(11, 6)B = (11,\, 6)

Step 1: xB=2(−3)−2=−6−2=−8x_B = 2(-3) - 2 = -6 - 2 = -8
Step 2: yB=2(0)−5=−5y_B = 2(0) - 5 = -5
Step 3: zB=2(4)−(−1)=8+1=9z_B = 2(4) - (-1) = 8 + 1 = 9
Step 4: Check each coordinate: 2+(−8)2=−3\tfrac{2 + (-8)}{2} = -3, 5+(−5)2=0\tfrac{5 + (-5)}{2} = 0, −1+92=4\tfrac{-1 + 9}{2} = 4 ✓
Answer: B=(−8, −5, 9)B = (-8,\, -5,\, 9)

Step 1: The centre is the midpoint of the diameter, so the endpoint formula applies
Step 2: xB=2(4)−(−1)=8+1=9x_B = 2(4) - (-1) = 8 + 1 = 9
Step 3: yB=2(−2)−5=−4−5=−9y_B = 2(-2) - 5 = -4 - 5 = -9
Step 4: Check: (−1+92,5+(−9)2)=(4,−2)\left(\tfrac{-1+9}{2}, \tfrac{5+(-9)}{2}\right) = (4, -2) ✓
Answer: The other endpoint is (9, −9)(9,\, -9)

常见问题

Given one endpoint A and the midpoint M, the other endpoint is B = (2x_M - x_A, 2y_M - y_A). It comes from solving the midpoint formula for B, and it extends to three dimensions by treating each coordinate the same way.

The midpoint formula averages two endpoints, so it adds and divides by 2. The endpoint formula reverses that: it doubles the midpoint and subtracts the known endpoint. They are inverse operations on the same relationship.

Not uniquely. One endpoint plus a distance describes a whole circle of possible endpoints. You need a direction, a slope, or a second constraint such as the midpoint to pin down a single answer.

Use the section formula. If P divides AB in ratio m:n from A, then B = A + ((m+n)/m)(P - A). Setting m = n = 1 gives back the ordinary endpoint formula B = 2P - A.

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