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Mathematics LibreTexts

11.4: Defect

( \newcommand{\kernel}{\mathrm{null}\,}\)

The defect of triangle ABC is defined as

defect(ABC):=π|ABC||BCA||CAB|.

Note that Theorem 11.3.1 states that the defect of any triangle in a neutral plane has to be nonnegative. According to Theorem 7.4.1, any triangle in the Euclidean plane has zero defect.

Exercise 11.4.1

Let ABC be a nondegenerate triangle in the neutral plane. Assume D lies between A and B. Show that

defect(ABC)=defect(ADC)+defect(DBC).

截屏2021-02-23 上午10.49.18.png

Hint

Note that |ADC|+|CDB|=π. Then apply the definition of the defect.

Exercise 11.4.2

Let ABC be a nondegenerate triangle in the neutral plane. Suppose X is the reflection of C across the midpoint M of [AB]. Show that

defect(ABC)=defect(AXC).

截屏2021-02-23 上午10.50.45.png

Hint

Show that AMXBMC. Apply Exercise 11.4.1 to ABC and AXC.

Exercise 11.4.3

Suppose that ABCD is a rectangle in a neutral plane; that is, ABCD is a quadrangle with all right angles. Show that AB=CD.

Hint

截屏2021-02-23 上午10.55.37.png

Show that B and D lie on the opposite sides of (AC). Conclude that

defect(ABC)+defect(CDA)=0.

Apply Theorem 11.4.1 to show that

defect(ABC)=defect(CDA=0

Use it to show that \meauredangleCAB=ACD and ACB=CAD. By ASA, ABCCDA, and, in particular, AB=CD.

(Alternatively, you may apply Exercise 11.3.1)

Advanced Exercise 11.4.4

Show that if a neutral plane has a rectangle, then all its triangles have zero defect.


This page titled 11.4: Defect is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.

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