Pitot theorem

The Pitot theorem in geometry states that in a tangential quadrilateral the two pairs of opposite sides have the same total length. It is named after French engineer Henri Pitot.^[1]

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Bernoullis equation can be used for a number of different situations. In this video we are going to apply it to what is known as a Pitot Static tube. This is an example of a Pitot static tube and the principle that it works on is that you want to measure a pressure difference between two points where one of the points is considered a stagnation point. A stagnation point is when the velocity equals 0. If we start with Bernoullis equation it says that P1 plus 1/2 times the density times V1^2 which is the velocity plus the specific weight times z1 equals the same thing P2 plus 1/2 time rho times V^2 plus gamma times z2. We can make some assumptions that make this a lot easier. If we say that z1 equals z2. In other words it is horizontal we also will say that 2 is going to be the stagnation point so V2 = 0. What we are now left with is that P1 plus 1/2*rho*V1^2 equals P2. We can use this equation either to solve for a difference in pressure or we can use this equation to solve for an actual pressure given the other pressure. Let's say that we have a plane and we have at the nose of the plane a pitot static tube. We have far away a point. What we want to do is find the pressure difference between point 1 and point 2. What do we know about this airplane. We are given that it is flying at 100 m/s, its elevation is 4000 m and we will us the expression we developed before. P2 equals P1 plus 1/2 *rho*V1^2. The simplest way to do this is that P2 minus P1 equals 1/2*rho*V1^2. We know that our V1 is 100 m/s. We can look up the density of air at at temperature of about minus 11 degrees C. This is 0.8194 kg/m^3. Our delta P is 1/2*0.8194*100^2 which is 4.1 kPa. We can also look up P1. If we look up the pressure of air at 4000 m we find that equals 6.2*10^4 Pa or 62 kPa.

Statement and converse

A tangential quadrilateral is usually defined as a convex quadrilateral for which all four sides are tangent to the same inscribed circle. Pitot's theorem states that, for these quadrilaterals, the two sums of lengths of opposite sides are the same. Both sums of lengths equal the semiperimeter of the quadrilateral.^[2]

The converse implication is also true: whenever a convex quadrilateral has pairs of opposite sides with the same sums of lengths, it has an inscribed circle. Therefore, this is an exact characterization: the tangential quadrilaterals are exactly the quadrilaterals with equal sums of opposite side lengths.^[2]

Proof idea

One way to prove the Pitot theorem is to divide the sides of any given tangential quadrilateral at the points where its inscribed circle touches each side. This divides the four sides into eight segments, between a vertex of the quadrilateral and a point of tangency with the circle. Any two of these segments that meet at the same vertex have the same length, forming a pair of equal-length segments. Any two opposite sides have one segment from each of these pairs. Therefore, the four segments in two opposite sides have the same lengths, and the same sum of lengths, as the four segments in the other two opposite sides.

History

Henri Pitot proved his theorem in 1725, whereas the converse was proved by the Swiss mathematician Jakob Steiner in 1846.^[2]

Generalization

Pitot's theorem generalizes to tangential $2n$ -gons, in which case the two sums of alternate sides are equal. The same proof idea applies.^[3]

References

^ Pritsker, Boris (2017), Geometrical Kaleidoscope, Dover Publications, p. 51, ISBN 9780486812410.
^ ^a ^b ^c Josefsson, Martin (2011), "More characterizations of tangential quadrilaterals" (PDF), Forum Geometricorum, 11: 65–82, MR 2877281. See in particular pp. 65–66.
^ de Villiers, Michael (1993), "A unifying generalization of Turnbull's theorem", International Journal of Mathematical Education in Science and Technology, 24 (2): 65–82, doi:10.1080/0020739930240204, MR 2877281.

External links

This page was last edited on 19 October 2023, at 14:24

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