If A, B, C, D, and E are collinear with B between A and C, C between B and D, and D between Cand E, and AC = BD – CE, then AB = BC = DE
To prove that AB = BC = DE, we will use the properties of collinearity and the fact that AC = BD – CE
To prove that AB = BC = DE, we will use the properties of collinearity and the fact that AC = BD – CE.
First, let’s consider the points A, B, C, D, and E on a line. We are given that B is between A and C, C is between B and D, and D is between C and E. This information tells us the order in which the points are arranged on the line.
Now, let’s use the property of collinearity. A, B, and C are collinear, so the distance AB + BC should be equal to the distance AC.
AC = AB + BC ——- (Equation 1)
We are also given that AC = BD – CE. Substituting this value in equation 1, we get:
BD – CE = AB + BC
Rearranging the terms, we have:
AB + BC = BD – CE ——- (Equation 2)
Now, let’s consider the points B, C, D, and E. We are given that C is between B and D, and D is between C and E. Using the property of collinearity, the distance BC + CD should be equal to the distance BD.
BD = BC + CD ——- (Equation 3)
We also know that AC = BD – CE, which means that BD = AC + CE. Substituting this value in equation 3, we get:
AC + CE = BC + CD
Rearranging the terms, we have:
AC + BC = CD + CE ——- (Equation 4)
Now, let’s combine equations 2 and 4:
AB + BC = BD – CE
AC + BC = CD + CE
Adding these two equations together, we get:
AB + 2BC + AC = BD – CE + CD + CE
Simplifying the equation, we have:
AB + 2BC + AC = BD + CD
Since D is between C and E, CD = DE. Rearranging the equation, we have:
AB + 2BC + AC = BD + DE
Since AB + BC = AC, we can substitute this value in the equation:
AB + BC + BC = BD + DE
Simplifying the equation, we get:
AB + 2BC = BD + DE
Since BD = BC + CD and CD = DE, we can substitute these values:
AB + 2BC = BC + BC + DE
Simplifying the equation, we have:
AB + 2BC = 2BC + DE
Subtracting 2BC from both sides, we get:
AB = DE
Therefore, we have proved that AB = DE.
To conclude, using the properties of collinearity and the given information AC = BD – CE, we have shown that AB = DE.
Additionally, we can recall that we proved earlier that AB + BC = AC. Since AB = DE, substituting this value in the equation gives us DE + BC = AC. Since AB = BC, we can also conclude that BC = DE.
Hence, AB = BC = DE.
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