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Midpoint

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Midpoint

The point on a line segment dividing it into two segments of equal length. The midpoint of a line segment is easy to locate by first constructing a lens using circular arcs, then connecting the cusps of the lens. The point where the cusp-connecting line intersects the segment is then the midpoint (Pedoe 1995, p. xii). It is more challenging to locate the midpoint using only a compass (i.e., a Mascheroni construction).

For the line segment AB in the plane determined by A=(x_1,y_1) and B=(x_2,y_2), the midpoint can be calculated as

M=(1/2(x_1+x_2),1/2(y_1+y_2)).
(1)

Similarly, for the line segment AB in space determined by A=(x_1,y_1,z_1) and B=(x_2,y_2,z_2), the midpoint can be calculated as

M=(1/2(x_1+x_2),1/2(y_1+y_2),1/2(z_1+z_2)).
(2)

In a right triangle, the midpoint of the hypotenuse is equidistant from the three polygon vertices (Dunham 1990).

TriangleMidpoints

In the figure above, the trilinear coordinates of the midpoints of the triangle sides are M_A=(0,c,b), M_B=(c,0,a), and M_C=(b,a,0).

The midpoint of a line segment with endpoints alpha_1:beta_1:gamma_1 and alpha_2:beta_2:gamma_2 given in trilinear coordinates is alpha:beta:gamma, where

alpha=2aalpha_1alpha_2+b(alpha_2beta_1+alpha_1beta_2)+c(alpha_2gamma_1+alpha_1gamma_2)
(3)
beta=a(alpha_2beta_1+alpha_1beta_2)+2bbeta_1beta_2+c(beta_2gamma_1+beta_1gamma_2)
(4)
gamma=a(alpha_2gamma_1+alpha_1gamma_2)+b(beta_2gamma_1+beta_1gamma_2)+2cgamma_1gamma_2
(5)

(left as an exercise in Kimberling 1998, p. 35, Ex. 15).

TriangleMidpointEq

Given a triangle DeltaA_1A_2A_3 with area Delta, locate the midpoints of the sides M_i. Now inscribe two triangles DeltaP_1P_2P_3 and DeltaQ_1Q_2Q_3 with polygon vertices P_i and Q_i placed so that P_iM_i^_=Q_iM_i^_. Then DeltaP_1P_2P_3 and DeltaQ_1Q_2Q_3 have equal areas

Delta_P=Delta_Q=Delta[1-((m_1)/(a_1)+(m_2)/(a_2)+(m_3)/(a_3))+(m_2m_2)/(a_2a_3)+(m_3m_1)/(a_3a_1)+(m_1m_2)/(a_1a_2)],
(6)

where a_i are the sides of the original triangle and m_i are the lengths of the triangle medians (Johnson 1929).

See also

Anticenter, Archimedes' Midpoint Theorem, Bimedian, Bisection, Brahmagupta's Theorem, Brocard Midpoint, Circle-Point Midpoint Theorem, Cleaver, Droz-Farny Theorem, Line Segment, Maltitude, Mascheroni Construction, Mediator, Multisection, Steiner Inellipse, Triangle Median Explore this topic in the MathWorld classroom

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More things to try:

  • line
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References

Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 120-121, 1990.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 80, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.

Referenced on Wolfram|Alpha

Midpoint

Cite this as:

Weisstein, Eric W. "Midpoint." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Midpoint.html

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