学懂，练会。Learn it today.下次接着来。Pick up tomorrow.

记得进度的 AI 学习 App。
用自己的资料练习、订正和复习。An AI study app that remembers your progress. Practise, get feedback and review with your own material.

下载 AppGet the App

把练习写进记忆MAKE IT STICK.

学习示例Study example

  1. 提出问题Ask why
  2. 一起推导Work it out
  3. 迁移练习Try it yourself

屋顶变尖，角的总和会变吗？A sharper roof. A different angle sum?

折一个纸帐篷：屋顶越尖，三个内角的和会不会变小？Fold a paper tent. If the roof becomes sharper, does the sum of its three angles change?

欧几里得在《几何原本》I.32中，把三个角的问题转成了一条直线的问题。我们沿着这个办法试一试。In Elements I.32, Euclid turned a question about three angles into a question about a straight line. Let’s follow that idea.

依据：《几何原本》I.32Source: Euclid, Elements I.32

把分散的角，放到一起看。Bring the three angles together.

ABC
  • 过A作一条与底边BC平行的线。Draw a line through A parallel to BC.
  • 两条边是截线：底角分别等于A旁边的内错角。The two sides are transversals. Each base angle equals its alternate angle beside A.
  • 加上A处原来的顶角，三个角正好铺在同一条直线上。Together with the original angle at A, the three angles fill one straight line.

为什么不能套用360°？Why doesn’t 360° apply?

360°描述绕同一点转一整周。三角形的三个内角却在三个不同顶点，不能把“一整周”直接套上去。360° describes one full turn around one point. A triangle’s interior angles belong to three different vertices, so a full turn cannot simply be substituted.

量几张纸上的三角形可以帮助发现规律，但有限次测量不是对所有三角形的证明。平行线给出了形状改变后仍成立的理由。Measuring a few paper triangles can suggest a pattern, but a few measurements do not prove it for every triangle. The parallel line explains why the result survives a change of shape.

180°，来自一个平角。180° comes from a straight angle.

在平面欧氏几何中，内错角相等，把两个底角移到顶点旁，再加上顶角，恰好组成180°。所以三角形的内角和不依赖它画得高、矮或斜。In a Euclidean plane, equal alternate angles place the two base angles beside the top angle. Together they form 180°, regardless of how tall, flat or tilted the triangle is.

α + β + γ = 180°

换个形状，方法还管用吗？Does the method still work on a different shape?

两个内角是50°和60°，第三角为什么是70°？试着说出依据。Two angles are 50° and 60°. Why is the third 70°? Explain the reason.

展开解法See the explanation

第三角 = 180° − 50° − 60° = 70°。减法用的是刚才用平行线证明的内角和，不是凭图形看起来的大小猜出来的。The third angle is 180° − 50° − 60° = 70°. The subtraction uses the angle sum we just proved with parallel lines, rather than a guess from the drawing.

问题的产生、推导的过程、方法的练习。Ask why. Reason it through. Put it to work.

问题的产生Where the question begins

推导的过程Follow the reasoning

方法的练习Put the method to work

苏格拉底式学习Socratic learning

从手头的内容学起。Start with what you’re studying.

带上你正在学的内容，在对话里提问、练习和订正。Bring what you’re studying. Ask questions, practise and work through feedback in the conversation.

套餐Plans

Plus

养成每日学习习惯A steady daily habit

¥19/ 月/ month

约 ¥190 / 年Approx. ¥190 / year

适合每天短时练习和随时追问。For short daily practice and follow-up questions.

  • 每月学习点数Monthly study points
  • 使用 Flash 和 Pro 模型Flash and Pro model usage

Pro

推荐Recommended

让学习更进一步Room to go deeper

¥39/ 月/ month

约 ¥390 / 年Approx. ¥390 / year

适合规律练习、详细反馈和持续学习。For regular training, detailed feedback and steady preparation.

  • 每月学习点数Monthly study points
  • 使用 Flash 和 Pro 模型Flash and Pro model usage

Max

专注高强度学习For intensive study

¥68/ 月/ month

约 ¥680 / 年Approx. ¥680 / year

适合频繁提问、密集练习和长时间学习。For frequent questions, intensive practice and longer study sessions.

  • 每月学习点数Monthly study points
  • 使用 Flash 和 Pro 模型Flash and Pro model usage

常见问题Frequently asked questions

开始使用Getting started

Yozi 适合谁？Who is Yozi for?
Yozi 面向从小学到成人备考的学习者，帮助你用手头的资料练习、理解反馈和安排复习。Yozi is for learners from primary school through adult exam preparation who want to practise with their material, understand feedback and keep up with review.
现在可以下载哪些平台？Which platforms are available?
下载 Mac 版即可开始，需要 macOS 14 或更新版本与 Apple 芯片。Download Yozi for Mac to get started. Requires macOS 14 or later and Apple silicon.
从哪里开始？How do I start?
下载应用、注册登录，在 AI 连接器中接入自己的 API，再选择可用型号。随后通过对话建立学习档案，开始练习。Download the app and sign in. Add your API connection in the AI connector, select an available model, then build your study profile through the conversation.
可以上传自己的资料吗？Can I use my own material?
可以。上传自己的学习资料，围绕资料提问与练习。模型使用哪些内容，仍按应用内的分享授权执行。Yes. Upload your study material and use it for questions and practice. Sharing with a model follows the permission you give in the app.

模型与费用Models & costs

BYOK 是什么？What does BYOK mean?
Bring Your Own Key，即自带 API Key。你选择 API 提供方、连接与型号；型号从所接 API 实时读取。Bring Your Own Key. You choose the API provider, connection and model. Available models are read from your connected API.
在哪里查看模型用量？Where can I view model usage?
在所选连接的提供方账户中查看模型用量。View model usage in your connected provider’s account.
Yozi 有订阅价格吗？Does Yozi have subscription prices?
Plus ¥19、Pro ¥39、Max ¥68。Plus ¥19, Pro ¥39 and Max ¥68.
型号不能用时会自动切换吗？Will Yozi switch models automatically?
不会自动换供应商或付费型号。当前型号不可用时会明确提示，由你选择切换。No. Yozi does not silently change providers or paid models. If a model is unavailable, you are told and can choose another one.

隐私与学习记录Privacy & study records

学习内容会发送到哪里？Where does my study content go?
使用模型需要把本次已授权的学习内容发送到你选择的 API 提供方。请同时查看该提供方的数据政策；不能把 BYOK 理解为所有内容只在本机处理。Model use sends the study content you have authorised to your chosen API provider. Review that provider’s data policy too. BYOK does not mean all processing stays on your device.
需要在官网填写密钥吗？Should I enter my key on this website?
不需要。官网没有 API Key 输入框。请在应用内的 AI 连接器中配置自己的连接。No. This website has no API key input. Configure your connection in the app’s AI connector.
这些示例是学生成绩吗？Are these real student results?
不是。网页展示的是学习过程示例与产品理念，不是学生成绩、用户评价或掌握准确率证据。No. The website shows illustrative learning examples and product principles. They are not student results, customer reviews or evidence of mastery prediction accuracy.
AI 批改是官方评分吗？Is AI feedback an official score?
不是。写作反馈属于 AI 评估，帮助理解与练习，不代表考试机构的官方评分。No. Writing feedback is an AI assessment to support learning and practice, not a score issued by an examination authority.

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