$\Delta BZM$ is a 30-60-90 triangle, so $PX = ZM = BZ / 2$, and symmetry gives $PZ = ZY / 2 = BZ / 2.$ Thus $\Delta PZX$ is an isosceles right triangle, and, similarly, so is $\Delta PYX.$ Thus the result is proved.
Recently a friend posed a Geometry problem to me: $\angle BAC = 120^{\circ}$, and $AX$ bisects $\angle BAC.$ If $BY$ and $CZ$ bisect angles $\angle ABX$ and $\angle XCA$, respectively, then $\angle ZXY$ is a right angle. After standard angle-chasing failed to produce an answer, I turned to my bag of problem solving techniques: Look at a Special Case; Investigate the Converse; and Ask a Related Question. Special Case: An easy way to simplify the problem is to assume that $\Delta ABC$ is isosceles. In this case, we can use special right triangles to prove our result. The symmetry of $\Delta ABC$ shows that $ZY || BC \perp AX.$ Because $BY$ is an angle bisector, $\angle ZBY = \angle YBX = 15^{\circ}.$ This shows that $\Delta ZBY$ is isosceles with base $BY$. As a result, $BZ = ZY.$ $\Delta BZM$ is a 30-60-90 triangle, so $PX = ZM = BZ / 2$, and symmetry gives $PZ = ZY / 2 = BZ / 2.$ Thus $\Delta PZX$ is an isosceles right triangle, and, similarly, so is $\Delta PYX.$ Thus the result is proved. The Converse makes the right angle the assumed and attempts to prove that two segments are angle bisectors. $\angle BAC = 120^{\circ}$, and $AX$ bisects $\angle BAC.$ If $\angle ZXY$ is a right angle, then $BY$ and $CZ$ bisect angles $\angle ABX$ and $\angle XCA$, respectively. The converse is clearly proved false by the animation, because $\angle ABY$ grows and $\angle YBX$ shrinks. However, the animation lead to a related question, which I have not yet found a sufficient answer to. I am still investigating both the original and the related problems. Though I have not produced a complete answer, I do not believe I have wasted my efforts, and I've learned a lot along the way. I will take a break and return to the problem in a little while.
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Insofar as mathematics seeks exact answers, it limits the scope of the questions it considers. To answer non-mathematical questions, the mathematician must adapt their tools and expect inexact answers. I care about social questions of equity, fairness, and justice, and there is a history of mathematicians before me working to shed light on these situations. A common social question that mathematics can answer is the question of how to allocate some limited resource among several parties. A few examples of this type of social question are given below.
As mathematicians and lawmakers consider which and whether an algorithm is superior to a legislative or committee process, there are several pros and cons they keep in mind.
Recently, social scientists, lawmakers, and mathematicians have tried to apply mathematical reasoning to a different social problem. These researchers found a simple algorithm to determine whether an accused criminal should be held in custody while awaiting trial or released. To do so, the researchers articulated clear goals for the algorithm, used rigorous mathematics, and provided room for human judgement in the final decision-making. In so doing, they attempted to highlight the pros of algorithms while mitigating their cons. The Arnold Foundation outlined their methods and findings on their website.
The goals of the research are, “First, to determine the best predictors across jurisdictions of new criminal activity, failure to appear, and, for the first time, new violent criminal activity. Second, to develop a risk-assessment tool based on these predictors.” [3] Mathematics and statistics were used heavily. “Researchers started with 1.5 million cases drawn from more than 300 U.S. jurisdictions.” and “The study identified and tested hundreds of risk factors.” [3] An algorithm does not replace human judgement. “[It] is a decision-making tool for judges. It is not intended to, nor does it functionally, replace judicial discretion. Judges continue to be the stewards of our judicial system and the ultimate arbiters of the conditions that should apply to each defendant.” [4] I encourage the reader to learn more about the Arnold Foundation’s risk-assessment tool which is currently used in many jurisdictions in the United States. [1] NPR's Planet Money recently covered this algorithm's implementation in the state of New Jersey. The Arnold Foundation summarizes their development of the algorithm on their website as well as in two reports. [2] Website overview [3] More detailed report [4] Details of the algorithm |
About Me
I started this blog to share my transformation from math nerd to math nerd who loves to share math with young people. I teach high school in Hanoi, Vietnam. Your comments are always welcome. Archives
May 2021
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