A sample layout with a tabbar in the left middle of the page.

middle_tabbar_layout.dart

import 'package:flutter/material.dart';
import 'package:flutter/src/widgets/ticker_provider.dart';

void main() => runApp(MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      title: 'Flutter Demo',
      home: MyHomePage(title: 'Flutter Demo Home Page'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  MyHomePage({Key key, this.title}) : super(key: key);

  final String title;

  @override
  _MyHomePageState createState() => _MyHomePageState();
}

class _MyHomePageState extends State<MyHomePage> with TickerProviderStateMixin {
  List<Tab> _tabs = [];
  TabController _tabController;

  @override
  void initState() {
    super.initState();

    _tabs = <Tab>[
      Tab(
        child: Text(
          'Lectures',
          overflow: TextOverflow.ellipsis,
        ),
      ),
      Tab(
        child: Text(
          'More',
          overflow: TextOverflow.ellipsis,
        ),
      ),
    ];
    _tabController = TabController(length: _tabs.length, vsync: this);
  }

  @override
  void dispose() {
    _tabController.dispose();
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    MediaQueryData mqd = MediaQuery.of(context);
    double imageHeight = mqd.size.height * 2 / 5;
    double tabBarHeight = mqd.size.height * 1 / 10;
    Color tabColor = Colors.white;
    return Scaffold(
      appBar: PreferredSize(
        preferredSize: Size.fromHeight(
          imageHeight + tabBarHeight,
        ),
        child: Container(
          padding: mqd.viewPadding.copyWith(bottom: 0),
          color: Colors.cyan,
          child: Column(
            children: <Widget>[
              Placeholder(
                fallbackHeight: imageHeight,
              ),
              Row(
                children: <Widget>[
                  Container(
                    width: mqd.size.width * 1 / 2,
                    height: tabBarHeight,
                    color: tabColor,
                    child: TabBar(
                      indicatorColor: Colors.deepPurple,
                      labelColor: Colors.black,
                      unselectedLabelColor: Colors.grey,
                      indicatorWeight: 3,
                      tabs: _tabs,
                      controller: _tabController,
                    ),
                  ),
                  Container(
                    width: mqd.size.width / 2,
                    height: tabBarHeight,
                    color: tabColor,
                    alignment: Alignment.centerRight,
                    padding: EdgeInsets.only(right: 20),
                    child: IconButton(
                      icon: Icon(Icons.arrow_drop_down_circle),
                      color: Colors.grey,
                      onPressed: () {
                        print('Down arrow pressed');
                      },
                    ),
                  ),
                ],
              ),
            ],
          ),
        ),
      ),
      body: TabBarView(
        children: <Widget>[
          Container(color: Colors.green),
          Container(color: Colors.yellow),
        ],
        controller: _tabController,
      ),
    );
  }
}